Minimum Chromaticity and Efficient Domination of

Circulant Graphs

Goran Ruzic B.Sc. Simon Fraser University 200 1

A THESIS SUBMITTED IN PARTIAL FULFILMENT

OF THE REQUIREMENTS OF THE DEGREE OF

In the School of

Computing Science

O Goran Ruzic 2003 SIMON FRASER UNIVERSITY

November 2003

All rights reserved. This work may not be reproduced, in whole or in part, by photocopy

or other means without permission of the author.

Approval

Name: Goran Ruzic

Degree: Master of Science

Title of thesis: Minimum Chromaticity and Efficient Domination of Circulant

Graphs

Examining Committee: Dr. Valentine Kabanets

Chair

.. 34- - --- ,-- ~ r f l & e ~ h deters, Senior Supervisor

-- r - - Dr. Petra ~erendrink, Supervisor

-

Dr. Ladislav Stacho, Examiner

SIMON FRASER UNIVERSITY

PARTIAL COPYRIGHT LICENSE

I hereby grant to Simon Fraser University the right to lend my thesis, project and extended essay (the title of which is shown below) to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. I further agree that permission for multiple copying of this work for scholarly purposes may be granted by me or the Dean of Graduate Studies. It is understood that copying or publication of this work for financial gain shall not be allowed without my written permission.

Title of Thesis/Project/Extended Essay:

Minimum Chromaticity and Efficient Domination of Circulant Graphs

v

Author: / v .' - \ I

(signature)

Goran Ruzic

(name)

(date) t

Abstract

Effective solutions to problems encountered in networks are often based on

whether the elementary set can be partitioned into classes according to some specific

criteria. The chromatic number of a graph G(V,E) is the minimum number of colours

needed to colour the vertices of G such that no two adjacent vertices have the same

colour. An independent set D of G is called an efficient dominating set of G if every

vertex not in D is adjacent to exactly one vertex in D. The circulant graph G = GC(n;S) of

order n is a graph on vertex set V= (vO, vl, ..., v ~ - ~ } and connection set S (1,2 ,..., 19 with an edge joining vi and vj whenever i= (j + sk) mod n, s k ~ S. In this thesis, we show

that many of the circulant graphs which are often proposed as underlying architectures

for computer networks are at most three colorable. Moreover, we establish infinite

families of GC(n;S), ISI=2 with chromatic number at most 3. We provide a complete

characterization of circulant graphs GC(n;S), ISI=2 which admit an efficient dominating

set.

Acknowledgments

I would like to thank my supervisor, Dr. Joseph Peters, for his exceptional

guidance, support and warm friendship. He helped me learn the foundations of research

and intellectual collaboration. I am also thankful to my committee members, Dr. Petra

Berenbrink and Dr. Ladislav Stacho, for providing valuable feedback on the ideas

presented in this thesis.

I am especially grateful to my family, in particular, my parents and brother for

their never-ending love and understanding.

Contents

Approval

Abstract

Acknowledgments

Contents

List of Figures

1 Introduction

1.1 Properties of Circulant Graphs ...................................................... . . 1.2 Problem Definition .......................................................................................

1.2.1 Vertex Colouring ......................................................... 1.2.2 Efficient Domination .....................................................

1.3 Related Work ........................................................................ 1.3.1 Minimum Graph Colouring .............................................

1.3.1.1 Integer Distance Graphs ..................................... 1.3.2 Efficient Domination ..................................................... . . ......................................................................... 1.4 Contribution

1.5 Plan ....................................................................................

2 Colouring Circulant Graphs

.......................................................................... 2.1 Introduction ......................................... 2.2 Circulant Graphs with k Chord Lengths

..................................... 2.3 Circulant Graphs with Two Chord Lengths

3 Efficient Domination in Circulant Graphs

.......................................................................... 3.1 Introduction ............................................... 3.2 Circulant Graphs with Two Chord Lengths

....................................................... 4 Conclusions and Open Problems

References ......................................................................................

List of Figures

3-proper and 2-proper colourings of hypercube Q3 .........................................

Perfect and not independent dominating set of G ..........................................

Independent and not perfect dominating set of G .........................................

Efficient dominating set of G .................................................................

3-clique {ai, aj. ak) in G and N[O] defined by {ai. aj. ak) .................................

Graph G' induced by vertices sl. s2. s3..sl. -s2 and .s3 ....................................

The graph induced by G 'n (0) ...............................................................

Proper colouring of Gc(3; S={l)) and intermediate colouring of

Gc(12; S=(1.4)) ...................................................................................

Proper colouring of Gc(l 2; S={ 1 .4)) ........................................................

Proper colouring of GC(48; S={l .4. 16)) ....................................................

Pattern definition ..............................................................................

Proper 3-colouring of GC(12; (3.4)) .........................................................

Reorganization of the colouring pattern for Gc(l 8; (3.4)) ................................

Definitions of the patterns P1 and P2 for Sz(3.4.5) ........................................

............................................. Proper 3-colouring of Gc(n; (3.4. 5)) 3 85 n545

C010~ring scheme for GC(n=lYbl+lYebl=49; {sl=8 . s2=1 7)) ...............................

Construction of Ymb from Yb for s1=8 and s2=28 ...........................................

.................................... Hypercube Q3 as a covering of the complete graph Kq

.......................... Unobtainable construction of D, with 2 members at distance 3

.......................... Unobtainable construction of D, with 2 members at distance 4

2 1 Graphical representation of G=Gc(42; (6. 1 1 }) ............................................. 47

22 Periodic behavior pattern of the members of D. ............................................ 49

23 Graphical representation of G for a = 4 (mod 5) ........................................... 51

24 Graphical representation of G for a = 3 (mod 5) ............................................ 52

25 Graphical representation of G for a = 2 (mod 5) ............................................ 53

26 Graphical representation of G for a = 1 (mod 5) ............................................ 54

27 G=Gc(20; (3. 6)) as a covering of K5 ......................................................... 56

vii

1 Introduction

The analysis and design of interconnection networks is motivated by recent

developments in technologies such as optical fiber and by progress in parallel and

distributed computing. Engineers and computer scientists often use graphs to model the

topological structure of an interconnection network. Switches, processing elements, or

memory modules correspond to the vertices of the graph while communication links

correspond to the edges. Graph theory is an effective tool for solutions to problems often

encountered in networks. In this thesis, circulant graphs are used to model

interconnection networks and the problems we study are minimum vertex colouring and

efficient domination on circulant graphs.

The design of an interconnection network should conform to the basic principles

summarized in [42]: small and fixed degree of each node in the network, small

transmission cost, maximum fault tolerance, easy routing algorithms, embeddability of

other topologies, symmetry, extendability, and efficient layout of VLSI circuits. Due to

the cost and engineering limitations, the maximum node degree is a primary constraint in

network design. Most of the popular network topologies have the drawback that the size

of the network cannot be increased incrementally. Instead, the size is limited to a multiple

of some factor and the number of connections to each node is proportional to the growth

of the network. However, circulant graphs, a family of Cayley graphs, allow for

incremental extendability, with the number of connections to each node and the diameter

remaining constant in the networks they model. Cayley graphs are often suggested as

models for interconnection networks because of their high symmetry. Since these graphs

are vertex-transitive they allow for the development of efficient routing algorithms.

(Informally, a network is vertex transitive if the view of its structure is the same from

every node.) In the case of a prime number of vertices, circulants are known to be the

only vertex-transitive graphs. For these reasons, circulant graphs have received a lot of

attention in recent years.

The graph colouring problem is one of the oldest problems in graph theory. In

fact, the most famous graph colouring problem was conjectured in the nineteenth century.

The problem known as map colouring is to find the minimum number of colours required

to colour countries on a world map so that no two adjacent countries (or regions) share

the same colour. The conjecture that at most four colours are necessary was finally

proved in 1977 [2,3]. In recent decades, many problems of practical interest have been

related to graph colouring. For example, time tabling, scheduling, printed circuit board

testing, and frequency assignment can all be modeled as graph colouring problems. In

general, an application involves the construction of a graph with vertices representing

points of interest. An edge of the graph connects two incompatible points of interest. A

colouring problem is to assign a colour to each vertex such that incompatible points are

differently coloured. The most often cited examples of applications of graph colouring

are time tabling and scheduling problems. Scheduling problems often require assurance

that a number of painvise restrictions on which jobs can be done simultaneously are

respected with minimum cost. A typical example is the "finals scheduling problem":

Devise a timetable of final examinations so that no student has two examinations

scheduled at the same time. The problem of determining the minimum number of time

slots with respect to the given constraint can be modeled as a graph colouring problem.

The problem of testing printed circuit boards for short circuits has been studied in [IS].

The problem can be modeled as a graph in which the vertices correspond to the nets (i.e.

circuits) on the board and there is an edge between two vertices if there is a potential for a

short circuit between the corresponding nets. Colouring the graph corresponds to

partitioning set of the nets into supernets, where the nets in each supernet can be

simultaneously tested for shorts against all other nets, thereby speeding up the testing

process [40]. An application of graph colouring for computer networks has recently been

suggested by Michael T. Goodrich. In fact, in [20] he devises a concept which utilizes the

minimum chromaticity of an underlying architecture as an alternative to a hashing

h c t i o n required for verification in a "secure routing" algorithm. To the best of our

knowledge, this is the first application of graph colouring to a one-to-one communication

scheme on a specific network topology.

A number of questions related to the concept of domination come from

applications of graph theory. The application that is the most often cited in the literature

is the design of efficient topologies for interconnection networks of high performance

parallel computers. There are many different ways to define the efficiency of a computer

network (diameter, reliability, etc). One such definition considers a network topology to

be efficient if it guarantees fair access to a limited number of resources (shared memory,

I10 devices) to each processor in the network. Yet another application deals with the

placement of software packages, such as code libraries, at individual processor nodes in a

computer network. If the software package needs to be installed on every node in a

network, the total expense of the design becomes undesirable. A nayve solution to the

problem is to ensure that each processor in the network has a short path to the resource.

However, the shortest paths are not necessarily the most desirable paths. Difficulties may

arise because each link in the network can be assigned a different weight according to its

bandwidth or some similar criterion. A better solution to the problem of the allocation of

the libraries across the network can be achieved by considering a minimum dominating

set. Moreover, it is not hard to see that the most efficient solution is the one that avoids

overlap. Therefore, the allocation problem can be solved by finding a more restrictive

version of a minimum dominating set, namely an independent perfect (i.e. efficient)

dominating set. Other researchers [34] have proposed a model for a broadcast routine in

wormhole-routed networks that uses the concept of a dominating set. They observed that

several collective communication routines (i.e. multicast, all-to-all broadcast) in mesh-

like systems [34] benefit from the use of dominating sets. The following protocol has

been suggested: the first step of the broadcast is a multicast from the source node to the

set of dominating nodes in the network. The second step is message dissemination from

the dominating nodes to their neighbours.

1.1 Properties of Circulant Graphs

Definition 1.1

Let n E Z+ and let the connection set of chords be S={sl,sz,. . ., sk} with S c {1,2,. . .,

141 }. The circulant graph G=GC(n;S) of order n is a graph on vertex set V={O, 1,. . ., n-

1 } with an edge joining i and j whenever i=(j+ sk) (mod n), s k E S.

A geometric circulant graph G = GGC(n; 4 is a circulant graph with S = ( d o ,

dl ,.. ., d m )for 1 < d 5 and m satisfying d m +1 < n 5 dm" +l. A recursive circulant 14 graph RGc(n; d) is a geometric circulant graph with order n = c. d m , 1 < c5 d [36].

Theoretical properties of circulant graphs have been studied extensively and are

surveyed in [5]. Most of the earlier research concentrated on either determining the value

of the diameter D for a given n and arbitrary set S, or determining the maximum value of

n for given D and arbitrary S. In recent years, research regarding circulant graphs has

expanded to areas such as recognition, hardness, spectral properties, isomorphism,

enumeration, and Hamiltonicity [6, 11, 32, 361. It has been conjectured in [I] that every

connected Cayley graph on an abelian group admits a Hamiltonian decomposition. The

circulant graphs with two chord lengths (i.e. double loop graphs) and recursive circulant

graphs have been proven to be decomposable into Hamiltonian cycles [6, 81. A circulant

graph G=Gc(n;S) is deemed optimal if the set S yields a minimum value of D for given n

and k. The problem of constructing optimal circulant graphs, even for k=2 and arbitrary

sl, s 2 turns out to be difficult. The exact value of D for a given n has been obtained only

for specific values of sl,s2, ..., s k when k=2 and k=3 [7], and there is no known efficient

algorithm which determines the values of n for which optimal circulant graphs can be

constructed for k2 4 [5]. The class GC(n; (sl = D, s 2 = D+l)), 2~~ -2D + 1 < n 5 2~~ + 2D + 1 is one of the few optimal cases for k 2 . Algorithms for some problems, such as

finding shortest paths and determining the diameter, which take polynomial time on

arbitrary graphs may exhibit exponential behaviour on circulant graphs [9]. This is

because GC(n;S) has a very compact representation consisting of only k=(SI integers (in

addition to the order n).

Network topologies modeled by recursive circulant graphs [37] show better

performance in terms of network parameters such as diameter, node visit ratio and

internode distance compared to networks modeled by hypercubes with the same number

of nodes and communication links. The double loop graph GC(n; {sl = D, s 2 = D+1)) has

been extensively studied as a model for interconnection networks in [7,10,29,38].

1.2 Problem Definition

In this section, we give formal definitions of graph homomorphism, vertex

colouring and dominating set. Following the set of definitions, we give detailed

examples.

Vertex colouring and efficient domination in circulant graphs can be related to

graph homomorphism.

Definition 1.2

Let G(V,E) and H(V,E) be graphs. A homomorphism from G to H is a mapping h: V(G)

+ V(H) such that for each edge (x, y) of G, (h (x), h 0) is an edge of H.

The relationship between efficient dominating sets and a special case of graph

homomorphism is shown in Chapter 3.

1.2.1 Vertex Colouring

Definition 1.3

Let G be a graph with vertex set V(G) and edge set E(G). An a-colouring of a graph G is

a labelling C:V(G)+ A, where IAl=a. The labels are colours; the vertices of one colour

form a colour class. An a-colouring is proper if adjacent vertices have different labels. A

graph is a-colourable if it has a proper a-colouring. The chromatic number X (G) is the

least a such that G is a-colourable[43].

Note that graph colouring is a special case of graph homomorphism. In fact, a

homomorphism from G to a complete graph K, is equivalent to an a-colouring of G.

Thus, graph homomorphism is a generalization of colouring.

Example 1.1

Let G be a hypercube on 8 vertices. A proper 3-colouring of G is depicted on the left of

Figure 1. The colouring is proper because any two adjacent vertices are in different

colow classes. A proper 2-colouring of G is shown on the right of Figure 1. X (G)=2.

Figure 1 : 3-proper and 2-proper colourings of hypercube Q3.

1.2.2 Efficient Domination

Definition 1.4

Let G be graph with vertex set V(G) and edge set E(C). A subset D of vertices is called a:

dominating set of G if 'du EV(G)\D 3 v ~ D such that u and v are adjacent;

perfect dominating set of G if b'u EV(G)\D 3 ! v ~ D such that u and v are adjacent;

efficient dominating set or independent prefect dominating set of G if 'du

EV(G)D 3!vED such that u and v are adjacent and D is an independent set of G.

The domination number y(G) is the minimum size of a dominating set in G.

We say that vertex v is a dominator if VED. In the remainder of this chapter, we use

D, to denote an efficient dominating set of G.

Example 1.2

Let G(V,E) be a path of length 3. Since no vertex has a degree greater than 2, then

any dominating set D contains at least 2 vertices. The instances that follow show the

relationships among different types of minimum dominating sets of G. Vertices that are

members of the dominating set D are coloured gray.

The dominating set D shown in Figure 2 is perfect but is not an independent

dominating set because every vertex in V(G)\D has exactly one neighbor in D but the

members of D are not independent in G.

Figure 2: Perfect and not independent dominating set of G.

The dominating set D shown in Figure 3 is independent but is not a perfect

dominating set because the members of D have a neighbor in common.

Figure 3: Independent and not perfect dominating set of G.

The dominating set D shown in Figure 4 is an efficient dominating set of G

because the members of D are independent in G and have no neighbor in common. Note

that no other dominating set of G satisfies the efficiency property.

Figure 4: Efficient dominating set of G.

1.3 Related Work

In this section we give a literature survey on problems related to minimum graph

colouring and efficient domination of circulant graphs.

1.3.1 Minimum Graph Colouring

The highly regular structure of circulant graphs gives the impression that the

problems of computing the basic graph parameters x(GC), size of the maximum

independent set a(GC), and size of the maximum clique w(Gc) are not as hard as for a

general graph. However, it is shown in [l 11 that MAXIMUM CLIQUE and MINIMUM

GRAPH COLOURING are NP-hard when restricted to circulant graphs. The complexity

of MAXIMUM INDEPENDENT SET is also NP-hard, and it follows as a consequence

from properties of GC (discussed in Chapter 2). The chromatic number of a circulant

graph with the connection set S=(l, s ) is studied in [17]. They have obtained various

results specific to the relationship between the chord length s and the order of Gc.

Spectral properties of circulant matrices are used in [ l l ] in order to obtain proper

colourings of GC and to give an upper bound on the chromatic number.

1.3.1.1 Integer Distance Graphs

The minimum colouring of circulant graphs has been related to the chromatic

number of integer distance graphs in [44].

Definition 1.5

The integer distance graph G~(s) is the graph with vertex set Z , and connection set

S c N . Two vertices u and v are adjacent iff lu-vl E S.

The problem of determining the minimum colouring of G~(s) was motivated by the well-

known Euclidean plane colouring problem: What is the smallest number of colours

required to colour the points of the Euclidean plane such that any two points at unit

distance have a different colour? It is known that the answer is between four and seven

[13, 351, while the exact number of colours is still an open problem. Minimum colouring

of Gd(S) is an extension of the 1-dimensional version of the problem. Integer distance

graphs were first introduced in [16]. The chromatic number of Gd(S) was studied in [15,

16,441.

The existence of a minimum colouring of an integer distance graph with the finite

set S implies the existence of a periodic minimum colouring [16]. (A colouring C:

Z + ( ~ 1 ~ ~ 2 , . . .,ck) is called periodic with period p if C(v)=C(v+p), p E N .) To see this

fact, partition the integer line into segments of length z+l, z=max(x: X E S) and assume

that Gd(S) is properly a-coloured by some function C. Since there are finitely many

different ways to colour a single segment (namely a"" different patterns) and the integer

line has an infinite number of segments, it follows that there exist two segments [uo, uZ]

and [vo, v,], vo> uo coloured with the same colouring pattern by function C. Then, Gd(S)

can be properly re-coloured with a colours with the pattern defined by segment [uo, vo-1]

and period p=vo-uo . In order to show the relationship between integer distance graphs and circulant

graphs, we restrict our attention to a finite connection set S. The existence of X(Gd(S))

with period p implies X(G~((~.~;S))=~(G~(S)) , t~ Z'. However, z(GC(n;S)) with

t.p+lln<(t+l).p, remains specific to circulant graphs. For example, let S={sl,sz,. . . ,sk) , be

a finite set of positive odd integers. Gd(S) is a bipartite graph with vertex sets V1 and V2,

containing odd and even integers respectively. In other words, X(Gd(s))=2 with period

p=2, which implies X(GC(n;S))=2 for any even n. However, as we show in Corollary 2.1,

X(Gc(n;S))23 for n odd. In general, for an arbitrary connection set S, the chromatic

number of Gd(S) gives a lower bound for X(Gc(n;S)).

1.3.2 Efficient Domination

More often than not, a solution to a dominating set problem has been

approximated by either a greedy algorithm or a branch and bound algorithm. However,

graphs with regular structure such as circulants and Cartesian product graphs allow for

the implementation of algorithms based on dynamic programming [30,39]. Solutions

obtained by dynamic programming techniques rely on periodic behavior of the

dominating set. For example, dynamic programming algorithms [30,39] for y(P,xPr) (i.e.

for an nxr grid) aim to identifl periodicity, and once the period is determined, induction

gives y(PnxPr) for a given n and any r. Finite state spaces are used in [30]. Min-max

algebra and the minimum domination of circulant graphs are explored in [39]. Moreover,

A. Spalding at al. [39] obtained a formula for the domination number y of a circulant

graph GC(n;S) as a function of n and SG (1,2,. . .,9).

The related problem of determining sufficient and necessary conditions for a

graph to admit an efficient (i.e. independent perfect) domination, due to its much more

restrictive nature, has been solved exactly for many regular graph structures such as

meshes and tori [21]. (In [21], the authors also give an algorithm for determining the

number of independent perfect dominating sets in trees.) Necessary conditions for the

existence of efficient dominating sets in selected examples of cross-product graphs are

presented in [33]. Livingston and Stout [31] completely characterize the existence of

efficient dominating sets for hypercubes and cube connected paths, and give the structure

of all such sets. However, it is important to point out that the characterization of efficient

domination for de Bruijn graphs and cube-connected cycles given in [31] is incomplete.

Dejter and Serra [14] explored families of nested Cayley graphs each of which has at

least one efficient dominating set. Recently, Lee showed that the Cayley graph for an

abelian group admits efficient domination if and only if it is a covering graph of a

complete graph [28]. In Theorem 3.3, we give an alternative proof that a circulant graph

admits efficient domination if and only if it is a covering graph of a complete graph.

1.4 Contributions

We determine families of circulant graphs for which each graph G = GC(n; S) has

Z(G)53. In particular, we show that there exists an no such that V n a o X(G)<3 r 7

s2#2.s1). We prove that recursive circulant graphs with df2 are at most 3-colourable. We

also provide a complete characterization of circulant graphs GC(n;S), ISI=2 which admit

an efficient dominating set.

1.5 Plan

Chapter 2 explores colouring schemes and hardness results on circulant graphs.

Chapter 3 deals with sufficient and necessary conditions for a circulant graph with two

chord lengths to admit efficient domination, as well as with the relationship between

efficient dominating sets and a special case of graph homomorphism. In the introduction

sections of both chapters we provide improvements and/or alternative proofs to known

results for the problems.

2 Colouring Circulant Graphs

In this chapter, we study the chromatic number of circulant graphs and determine

families of circulant graphs for which each graph GC(n;S) is at most 3-colourable.

2.1 Introduction

In this section, we focus on the computational complexity of the chromatic

number for circulant graphs. For the completeness of this thesis, we present proofs of the

hardness results. Our proofs are alternative and informal so they are accessible to a more

general reader. In Lemma 2.1, we give a better bound than the one given in Lemma 2 in

[ll]. The bound concerns the relationship between the independence number and the

chromatic number. Theorem 2.1 and Theorem 2.2 deal with the computational

complexity of w(GC) and x (GC) respectively. Before proceeding any further, we state the

following fact proved in Lemma 1 in [ l 11.

Fact 2.1

For any n, there exists a sequence of nonnegative numbers ao, al,...,a,,-1 which are

distinct modulo 8r10g"1, such that:

all sums ai+aj are distinct modulo 8r10gn1 -1,

all differences a,-a, are distinct modulo 8r10gn1 -1,

all sums ai+aj+ak are distinct modulo 8r'0gn1 - 1.

Moreover, the sequence ao, al,...,a,-1 is computable in polynomial time and the claim

remains true modulo any integer m satisfying m>3.( 8r10g"1 -2).

Theorem 2.1

MAXIMUM CLIQUE restricted to circulant graphs is NP-hard.

Proof:

From an arbitrary graph G, we construct a circulant graph GC such that the

maximum cliques in G and Gc are of the same size. Due to the vertex transitivity of

circulant graphs, if there is a clique of size k in GC, there must also be a clique of size k

contained in N[O] in Gc. So, the maximum cliques in Gc and N[O] are of the same size.

Consider a circulant graph GC with grbgnl -1 vertices labelled 0, 1,. . . , 8r10gn1 -2 and an

arbitrary graph G with n vertices, labelled ao, al, . . ., a,-1 such that the sequence ao, al,. . .,

a,-1 satisfies the properties stated in Fact 2.1. (Throughout the remainder of this proof we

assume that all arithmetic with respect to ao, al,. . ., a,-1 and vertices of Gc is carried out

mod (grIogn1 -l).) Let the connection set S of Gc contain the chord lengths + (arayl

whenever there is an edge (a,, a,) in G. So, if there exists a 3-clique (ai, aj, ak) in G

(shown on the left of Figure 5), then chord lengths + lai - ajl, + lai - akl, + laj - akl are in

the connection set S of GC.

Figure 5: 3-clique {ai, aj, ak) in G and N[O] defined by {ai, aj, ak).

Without loss of generality let sl= ai - aj, s2= ai - ak and s3= a, - ak, as shown on

the right of Figure 5. (Remember that the vertices in the set N(0) are determined by the

chord lengths in the connection set S.) Define graph G' to be the graph induced by

vertices sl, $2, s~,-sI, -s2 and -s3. From the definition of the chord lengths s1, sz, s3, we can

derive the following equation: sl - s2 + s3 = 0. This equation can be rewritten in the

following six different ways: sl- s 2 = -s3, - s 2 + sl = -s3, sl+ s3 = sz, s3 + sl= s2, - s2+ s3 =

- sl, s3- s 2 = - s1. Each equation corresponds to one specific edge in G', as shown in

Figure 6. For example, equation sl- s2 = -s3 corresponds the edge (sl , -s3). The set

El={(sl, -s3), (-SZ, -s3),(s1, s2), ( ~ 3 , ~ 2 ) , (-SZ, -s1),(s3, -s1)) contains all six edges of G'.

Figure 6: Graph G' induced by vertices sl, s2, s3,-sl, -s2 and -s3.

The claim is that graph G' does not contain an edge outside El. To verify the

correctness of the claim consider vertices a, - a, and a, - a, E N(O), and an edge at - a,

joining these two vertices. Then,

According to Fact 2.1, the sums of three elements in the sequence ao, al,. . .,a,-1 are all

distinct modulo 8r'0gn1 -l), which implies that it must be the case that w=p, r=t, s=q or

r=p, t=q, s=w. Therefore, two vertices a, - a, , a, - as E N(0) are adjacent iff their

corresponding edges (a,, a,) and (a,, a,) are part of the same 3-clique in G. Consequently

G' has no edge outside El and the graph induced by the vertices of G' along with the

vertex 0 has a clique of maximum size 3 (see Figure 7).

Suppose that the clique K, V(K)={ul, 212,. . ., uk) is a clique of the maximum size

in G. Without loss of generality, consider the edges (ul , u2) and (ul , u3) in K. The edge

(u l , u2) forms a 3-clique with each vertex u3, ..., uk. By the argument in the previous

paragraph, each vertex from the set A12={u1 - u3, u1 - ~ 4 , . . ., u1- uk, ug - 242, u4 - 242,. . ., uk

- u2} E N(0) is adjacent to the vertex ul - u2 in GC. By the same argument, the set

AI3=(u1 - u2, u1 - u4 ,..., U I - uk, u2 - u3, uq - u3,.. ., uk - ug} contains the vertices ffom

N(0) that are adjacent to vertex ul - us in Gc. The cardinality of the set A=(ul - 242, ul -

u3} U (Al2 A13) is k and this gives an upper bound on the size of the maximum clique

in N(0). It follows by the argument above that the vertices in A12flAl3=(u1 - ~ 4 , . . ., u1 - uk } form a clique of size k-3 in GC. Hence, the vertices 0, ul - u2, ul - u3,. . ., ul - uk form

a clique of size k in N[O] and this is a maximum clique in Gc.

I Figure 7: The graph induced by G'U (0).

I Therefore, by computing the size w(GC) of the maximum clique in GC, we can

compute w(G), which is NP-hard.

t

It is important to note that a circulant graph GC has a clique of size k if the number

of vertices in GC is greater than 3.( 8r10gn1 -2) because the sequence ao, al,.. .,a,,.l maintains

its properties for n greater than 3.( 8r10gn1 -2) by Fact 2.1.

The maximum independent set in G equals the size of the maximum clique in its

complement graphG. Since every circulant graph has the property that its complement is

a circulant graph, then computing MAXIMUM INDEPENDENT SET in circulant graphs

is also NP-hard.

Lemma 2.1

Let G=GC(n, S) and integer r be such that gcd(r, n)=l and r<n. There exists a circulant

graph H (polynominally computable from G) of order n-r such that the size of the

maximum independent set in H is a(H)=min{r, a(G)). Moreover, x (H)=n when

r l a! (G), and x (H)>n when r>a(G).

Proof:

Let R={0,1,. . ., r-1), and let the vertex set of H be R x V(G). Two vertices (i,v), U,w)

E V(H) are adjacent if any of the following conditions is satisfied:

1. i=j,

2. F W ,

3. edge (v,w)€V(G).

Since V(H) is obtained by a Cartesian product of R and V(G), H can be graphically

represented with the vertices arranged into IRI=r rows and IV(G)I=n columns. We

demonstrate that H is a circulant graph by first re-labelling the vertices of H and then

showing that the resulting connection set SH satisfies the requirements of a circulant

graph. Let f be the re-labelling function of vertices of H, f (i,v)=i-n+v.r (mod n-r).

Throughout the reminder of this proof we assume that all arithmetic with respect to f is

carried out (mod n-r).

It follows from Property 1 of the definition of adjacency in H that vertices fiom

the same row are mutually adjacent. Without loss of generality, consider vertices (0, j),

Olj<n-1 (i.e. vertices positioned in row zero of H). According to the labelling function&

the vertices in the first row have labels 0, r, 2-r, . . .,(n-1)-r. Let X={r, 2-r, . . .,(n-l).r).

Then the edge connecting vertices (OJ) and (0,y) satisfies IAO, x) -AO, y)l=l x-r- y-rl=l(x-

y).rl EX. In fact, with respect to row i of H, the labels assigned to vertices in row i (i.e.

ien, i.n+r, i.n+2-r, ..., i-n+(n-1)~) are in the same class (mod r). The edge connecting

vertices (is) and (i,y) belongs to X because Hi, x)-Ai,y)l=l(i-n+x-r) - (i-n+y-r)l=l (x- y).r)l

Property 2 of the definition of adjacency in H implies that vertices in the same

column are mutually adjacent. Vertices in the jth column have labels j r , n+jr, 2-n+jr,

. . .,(r-l).n+j.r, and therefore belong to the same class (mod n). Let Y={n, 2.n, .. .,(r-l).n).

Then for any two vertices (x, j) and (y, j), the edge (Ax, j)-fi, j)l=l (x-n+j.r) - @n+j.r)l=l

(x- y).n)l is in the connection set Y.

Without loss of generality, consider (adjacent) vertices 0 and sl in G, S ~ E S.

According to Property 3 of the definition of adjacency in H, vertices (x,O) and @,sl),

O<x,y<r are adjacent. Let W1=(sl.r, n + sl-r, ...,( r-l).n + s1.r ). The labels of vertices

(x,O) and (y,sl) satisfy b , s1)- f(x,O)l=l0.n+sl-r) - x-nl=l@-x)w +sl.r)l E W 1. Also, for

any two vertices a and a+sl adjacent in G, vertices (x, a) and (y, a+sl) are adjacent in H

because their corresponding labels satisfy IfCv, a+sl)- Ax, a)l=l(y.n+(a+sl).r)-

(x.n+a.r)l=l*x)-n +sl.r)l E Wl. In general, for any two adjacent vertices a and a + ~ i in G,

SiES, vertices (x, a) and (y, a+si) satisfy IfCY, a+si)- Ax, a)l=l @n+(a+si)-r) -

Let SH=XUYUW1UW2U"'UWls1. It follows that SH is a proper connection set for a

circulant graph H.

We now continue proving properties of a (H) and x (H). Since vertices in the

same row of H form a clique, then vertices that are independent in H have to be in

different rows. Hence, the cardinality of the maximum independent set of H can never

exceed r, (i.e. a (H)lr). Let IG =(v1, v2, ...,%@)I be a maximum independent set in G.

Depending on the relationship between r and a (G), the following cases need to be

considered:

r< a (G). From the definition of adjacency in H, it follows that the set

IH=((O,vl), (1,v2), . . .(r-1, v,)) is an independent set in H. Moreover, IH is a

maximum independent set because the upper bound on a (H) is r.

r> a (G). Then IH=((O,~l), (1 ,v2), . . . (cx(G)- 1, v ~ ( ~ ) ) ) is an independent set

in H. It follows from the definition of adjacency in H that the existence of

an independent set in H with cardinality greater then lIHl contradicts the

fact that IG is a maximum independent set in G. Hence, IH is a maximum

independent set in H.

Therefore a (H)=min(r, a(G)) .

What is left to be shown is that x (H)=n when r< a (G), and x (H)>n when

r>n(G). Recall that for any graph H ' , I V(H ') I s x(H'). For r< a (G), H has n maximum a(H')

independent sets, namely ((O,vl+k), (l,v2+k), ...( r-1, v,+k)), k=0,1, ..., n-1. Since each

set can be assigned a different colour, H can be properly coloured with n colours.

Iv(H)I = r2.r - n - r - - -- n - r Moreover, --- -- - n implies x(H) =n. For r> a (G), the a(H) a(H) min(r,a(G)) r

- n - r - IV(H)I - n - r r chromatic number of H is greater than or equal to - n.-

a(H) a(G) '

Hence, x(H)> n.

Theorem 2.2

MINIMUM GRAPH COLOURING restricted to circulant graphs is NP-hard.

Proof:

Let G be an arbitrary graph on n vertices. We consider the following three

construction steps:

1. Using the construction method presented in Theorem 2.1, build a

circulant graph Gl(r) on n, vertices, for each r=3,4,. . ., n, such that n, >3.(

8r'0gn1 -2) and gcd(n,, r)=l. (Recall that a circulant graph Gl(r) has a

clique of size o(Gl(r))=o(G) if the number of vertices in Gl(r) is greater

than 3 .( 8F1"gn' -2).)

2. Using the fact that the complement of a circulant graph is a circulant

graph, build G2(r)= G,(r), for each r=3, 4 ,..., n. The maximum

independent sets of G2(r) satisfjr a(G2(r))=o(G1(r))=o(G).

3. Using the construction from Lemma 2.1, build a circulant graph G3(r)

from G2(r), for each r=3, 4, ..., n. (Note that the number of vertices in

G2(r) is n, and gcd(n,, r)=l .)

Suppose that there is a polynomial time Algorithm A which computes the

chromatic number of circulant graphs. By Lemma 2.1, for r in the range 3 I r 5 a(G2(r)),

Algorithm A gives x (G3(r))=nr, and for r in the range a(G2(r))<r 5 n Algorithm A gives

x (G3(r))> n,. By noting the maximum value of r for which x (G3(r))=nr, the maximum

independent set of G2(r) can be identified. Moreover, since a(G2(r))=o(Gl(r))=o(G),

MINIMUM GRAPH COLOURING restricted to circulant graphs is NP-hard.

2.2 Circulant Graphs with k Chord Lengths

In this section, we consider families of circulant graphs with k different lengths of

chords for kl. For certain families, we focus on the existence of no, such that every

circulant graph with n?no vertices is at most 3-colourable. In particular, in Theorem 2.3

we show that circulant graphs are at least 3-colourable if the number of vertices is odd.

The case when the connection set S contains only odd integers is covered by Theorem 2.4

and Theorem 2.5. In Theorem 2.6 we show that the number of vertices in Gc must be

divisible by 3 if GC is 3-colourable and contains a clique of size 3. The chromatic number

of geometric circulant graphs is shown to be at most 3 in Theorem 2.7. Finally, in

Theorem 2.8 we consider the minimum chromaticity of eirculant graphs for which

connection set S contains only consecutive chord lengths.

Theorem 2.3

Let G (V, E) be a simple d-regular graph, IldllVI-1. If the order of G is odd, then X (G)

23.

Proof:

Suppose that X (G) = 2. Since G is 2-colourable, G is a bipartite graph with

vertex partitions V1 and V2, where VlnV2=0 and IVlI+IV2I=IV(G)I=2.j+1, for some

j~ Z' . There are d-(Vil edges leaving partition V , i = 1, 2. Since the number of edges

entering and leaving each partition has to be equal it follows that d.(V1(=d-(V21. However,

IV1(=(V21 contradicts the odd order of G. I

Corollary 2.1

Let G=GC(n;S), Sf 0 and n=2.j+ 1, j E Z' . Then, X (G) 23.

Proof:

For Sf 0, G is a graph of regular degree greater than or equal to 1 and the result

follows immediately from Theorem 2.1. I

Theorem 2.4

Let G=GC(n; S), n=2-j, j~ Z', such that the connection set S contains only odd integers.

Then, X (G)= 2.

Proof:

Let Vo and V1 be two partitions of V(G) such that v€Vo if v=O (mod 2), and

v E V1 if v= 1 (mod 2), 'v'v E V(G). Consider vertices u and w such that u, w E Vo or u, w

E V1. In either case lu- wl (mod n) = 0 (mod 2) e S, because S contains no even integers.

Therefore G is a bipartite graph, and X (G)= 2.

I

Theorem 2.5

For the connection set S containing only odd integers, 3 no such that Vn> no , X (G=

GC(n;S)) 5 3.

Proof:

Let no'=3-sk, k=ISI, and s k >s(k-l) ...> sl>l. In the following, we derive colouring

function C: V(G)+{a, b, c). Let Pl=abab.. .a, P2=bcbc.. . b and P3=caca.. .c be colouring

patterns such that IPll=IP21=IP31=~k. Finally, let pattern P consist of PI, followed by P2,

ending with P3. The claim is that P properly colours G=GC(no';S).

Without loss of generality consider vertices u and w coloured by the same colour. There

are two cases to consider:

u and w belong to the same colouring pattern Pi, i=1,2,3. However, each colouring

pattern has the property that the colouring of vertices with respect to any colour is

periodic with period p=2. Since the connection set S contains no even integer,

then u and w cannot be adjacent.

u and w belong to the different colouring patterns. Without loss of generality let

U E PI, and W E P2. Using the same argument as the one presented in the previous

case, the neighbourhood of u defined by u + s,, V sic S cannot contain vertex w.

Moreover, the size of P3 assures that w cannot be in the neighbourhood of u

defined by u-si, V S. The remaining cases U E PI, w E P3 and u E P2, w E P3 are

similar and therefore omitted.

For any odd number of vertices satisfying n13.sk + 29, j~ Z' let P1=abab.. .a be the

colouring pattern such that IP1l= s k + 2.j. It is easy to see that pattern P properly colours

GC(n;S). Hence, for odd n> no'=3.sk, graph GC(n;S) is 3-colourable. (By Corollary 2.1, G

is coloured with the minimum number of colours because the order of G is odd.)

For any even number of vertices, GC(n;S) is 2-colourable by Theorem 2.4. Since

ne3-sk-l= no'-1 is even (recall that the connection set S contains only odd integers), then

'dn>nO X(G)53.

I

The result of the following theorem follows directly from Corollary 6.13.4 in [19]. We

include an alternative proof in order to emphasise the structure of the proof that we will

use for Theorem 2.9.

Theorem 2.6

Let G=GC(n;S) contain K3. If G is 3-colourable then 3 is a divisor of n.

Proof:

Since G contains a triangle, then it must be the case that 3 si,sj,sk€ S, such that

si+sJ=sk or si+sJ=n-sk. Without loss of generality let S~+SJ=S~. Let G be 3-coloured by

colouring function C, and consider a sub-graph of G, H= GC(n;S={ si, sj, s k 1). Suppose that gcd(n,sk)=1. Then all products msk, a=O,l,. . .n-1 belong to different

classes modulo n. Thus, H has a Hamiltonian cycle (0, si+sj, 2(s,+sj),. . ., (n-l)(si+sj)). We

show that the colouring function C is periodic along the Hamiltonian cycle with period 3.

Any colouring of the clique (0, si, si+sj) gives C(2si+sj)=C(0) because {si, si+sj, 2si+sj) is

a clique in H. Similarly,

Since H is 3-colourable then 3111. Similarly, if gcd(n,sk)# 1, then the order of every cycle

in H generated by the chord sk has to be divisible by 3. Since all such cycles are disjoint,

it must be the case that 3111. I

Theorem 2.7 Let G=RGC(n;d), n=c. d m , l<c<d and d#2. Then, X (G) I 3.

Proof:

Let Gi=GC(n=c. d' ; s={& , dl,. . . , d' )), PO. We derive colouring functions

Ci:V(G,)+{O, 1, 21, 20 . In order to obtain proper colourings of geometric circulant

graphs, we utilize the recursive structure and camider two cases: d even and d odd.

Case d even:

Go=GC(c;{l)) contains c vertices and a single chord. If c>2 then Go is a cycle of

length c. If Go contains only two vertices (i.e. c=2), we introduce an additional edge

connecting vertices 0 and 1 so Go is a cycle of length 2 (note that in this case Go is not a

simple graph, but this construction step is required in order to make use of the recursive

structure).

Let Co be an arbitrary, but proper colouring of Go and define Co'(u) = (Co(u) + 1)

(mod 3) and Co"(u)= (Co(u) +2) (mod 3), ' d u ~ v ( G ~ ) . Colourings Co' and Co"are proper

colourings of Go because two vertices coloured differently by Co (vertices that possibly

share an edge) are coloured differently by Co' and Co7'.

Figure 8: Proper colouring of GC(3; S={l )) and

intermediate colouring of GC(12; S=(l ,4)).

Consider an arbitrary vertex ueV(G1). Since G1 contains c-d vertices, vertex u can be

represented as u=x.d+y, where O w - 1 05x9-1. Let colouring C(u) be defined as C(u)=

Co(x). Then, all vertices of G1 are coloured properly with respect to the chord of length d,

but not with respect to the chord of length 1 (see the colowhg of G1 for c=3 and d=4

shown on the right of Figure 8). However, because Co(x) # Co'(x) # CoV(x) then

colouring pattern Co'(x) COw(x), 1 5 H - 2 ; properly colours vertices connected with the

chord of length 1 (see proper colouring of GI for c=3 and d=4 shown in Figure 9).

Figure 9: Proper colouring of Gc(12; S={l ,4)).

In fact, this idea can be extended to any level of recursion. 'v'u E V(Gi+l) 3 x,y such

that u=x.d+y, O w - 1 and the colouring function C,+l for Gi+l is defined as follows:

c, c4 y=O, d-1 (C,(x)+l) mod 3, y odd, y # d-1

(C, (x$+2) mod 3. y even, y + 0

To verify the correctness of the colouring of Gi+1 by Cj+l consider vertex

v E V(Gi+l) such that v=u+ d , lgSi+l. Since u=x-d+y and v= xed +y+ d j = (x+ d'-' ).d+y,

Ci+1 colours u and v according to the same remainder y. This ensures that Ci+l(~)# Cz+l(v)

because C, is a proper colouring of G,. The same fact can be observed by noticing that

vertices of G,+l in the same congruence class modulo d induce a sub-graph G,, and that G,

is properly coloured by either Cj(u) , C~'(U)= Ci (u)+l (mod 3) or C~"(U)= Ci (u)+l (mod

3), u E V(Gi). For vertices u,v E V(Gi+l) with v = u + do = u + 1, the definition of C,+l and

the fact that C, is a proper colouring of G, ensures that C,+i assigns different colours to

consecutive vertices in Gi+l. Thus, Ci+i is a proper colouring of Gi+l. In conclusion when

i=m, Gm=RGC(n;d), andX (G&X (Gm-i)= . . .=X (Go) L 3. A proper colouring of

G2=GC(48;(1, 4, 16)) is shown in Figure 10 (some chords are omitted to enhance the

overall clarity).

Figure 10: Proper colouring of GC(48; S={ 1 ,4, 16)).

Case d odd:

The minimum colouring of RGC(n;d) for odd values of d is covered by Theorem

2.1 (c even) and Theorem 2.2 (c odd) because the connection set S=( d o , dl ,..., d' )

contains only odd integers and the smallest odd value of d is 3. For consistency, we give

a recursive colouring function C,+l for Gi+1 assuming the existence of the proper

colouring of Gi by C,:

c,+, = {:c;., 1) mod 3, y even

y odd

The argument for the correctness of Ci+1 is similar to the case d even, so we omit

the details.

I.,

Corollary 2.2 Let G=RGC(n;2), n=2. dm .Then X (G) > 3.

Proof:

This follows immediately from Theorem 2.6 because G contains chords of length

and d=2.

Theorem 2.8

For connection set S={s1,s2,. . .ski sk >sk-l ... >sll l and sl2 21 and skf2sl ), 3 no such

that V n> no , X (G=GC(n;S)) I 3.

Proof:

First note that sl=l is only possible when e l . In this case, G is a cycle and

X (G) 5 3. We can assume in the remainder of this proof that sl>l. Our description of

the colouring function C: V(G)+ {a,b,c) consists of two cases: sk even and sk odd.

Case sk even:

Let pattern P consist of 3 blocks: 5 consecutive vertices coloured a, followed by 2

5 consecutive vertices coloured b, ending with % consecutive vertices coloured c. 2 2

Figure 1 1 : Pattern definition.

No two vertices coloured by the same colour by P are adjacent because the shortest chord

Let pattern PI consist of 3 blocks: Sk consecutive vertices coloured a, followed IT1

S has length s l > s . Thus P is a proper colouring of G if n=IPI. If n=j .IPI = j ( 3 - 1 ) for

2 2

some j>O, then the pattern Y , consisting of j repetitions of pattern P gives a proper 3-

colouring of G. The pattern Y j can be extended to values of n that are not multiples of

IPI by increasing the numbers of vertices in the blocks. In particular, the number of

vertices in any block of consecutive vertices with the same colour can be increased from

5 up to sl without violating the property that adjacent vertices have different colours. 2

Therefore, X ( G ) 4 for n s [j-(3. &), j(3.sl)l for any j>0. 2

The inequality j.(3.sl) t (j+l) . (3-5) is satisfied when j> s k

2 1 2-s, -sk 1 - S k S Setting no= :(3.$), we have V n t no, X ((34. 1 2.sl -'k 1

Case s k odd:

The main difference between the proof of this case and the s k even case is that we

use two colouring patterns instead of one. The two patterns complement each other and

together give better lower bounds than we could obtain with a single pattern.

consecutive vertices coloured b, ending with consecutive vertices coloured

c. Let pattern P2 consist of 3 blocks: - consecutive vertices coloured a, followed by 1;l consecutive vertices coloured b, ending with - consecutive vertices coloured c. 1;l

Finally, let pattern P consist of PI followed by P2.

There are s k vertices of each colour in P and no two vertices coloured with the

same colour inside P are adjacent because sl> - . Thus, P is a proper 3-colouring of G El

We define two patterns containing repetitions of P. The first pattern, Y, consists

of j>O repetitions of pattern P and gives a proper colouring when n=j.IPI= j . 3 ~ ~ . The

second pattern Y ', consists of j>O repetitions of pattern P followed by a single repetition

of PI and gives a proper colouring when n= j-IPI+IP1l= j.3sk + s k + - . Similar to the s k kl even case, we can extend the two patterns by increasing the numbers of vertices in the

blocks of consecutive vertices with the same colour from or up to sl without L l ls,l violating the property that adjacent vertices have different colours.

For nE [je(3'sk), ~ (6 - s l ) ] , j>0 Y, gives a proper 3-colouring of G. For nE [je(3.sk)

+sk+ ), j(6.sl)+ 2-sl+sl] , j>O Y ', gives a proper 3-colouring of G. Is, 1 The inequality j.(6.sl)? j.(3.sk)+ s k + is satisfied when j 2

3-(2-sl - sk )

Setting no' = 1 13:' ] (3- sk), we have 'v' n2 no7, X (G)<3. Similarly, 3-(2.sl - s, )

setting nd7 = *' - 1 (I%)+ sk+ I?], we have 'v' n> nu.', X (G)Y. 2.s, - Sk

Combining these two bounds, we get b' n> no = min{no', no"), X (G)<3.

Remark 2.1

The two bounds no' and no" that were derived in the proof of Theorem 2.6 for the s k odd

case work together to give better bounds. For example, for the graph

GC(n;S={28,29,. . .5 I)), no7=842 and no"=918. For the graph GC(n;S={29, 30, . . .5 I)),

no'=689 and n$=612.

Example 2.2

Based on the two cases in Theorem 2.8 (sk even and s k odd), we give the patterns for the

connection sets {3,4) and {3,4,5). Colouring of GC(12; {3,4)) consists of 2 repetitions of

pattern P as shown in Figure 12. The size of the each block in P can be increased by 1,

which gives proper colourings for GC(n; {3,4)) 131n118. The colouring cannot be

extended to n=19 in this way, but GC(18; {3,4)) can be recoloured, so that it consists of 3

repetitions of pattern P (see the right of Figure 13). Hence, GC(n; {3,4)) is 3-colourable

for n212. For the connection set S={3,4,5), we illustrate the definitions of patterns PI and

P2 in Figure 14. As shown in Figure 15, the sizes of 7 blocks can be increased by 1.

However, GC(45; {3,4,5)) can be recoloured, so that it ends with P2. Hence, GC(n;

(3,431) is 3-colourable for 1~238.

Figure 12: Proper 3-colouring of GC(12; {3,4)).

Figure 13 : Reorganization of the colomhg pattan for Gc(l 8; {3,4)).

Figure 14: Definitions of the patterns PI and P2 for S={3,4,5).

Figure 15: Proper 3-colouring of GC(n; {3,4,5)) 389145.

2.3 Circulant Graphs with Two Chord Lengths

In this section, we concentrate on circulant graphs with chords of two lengths. The result

in Theorem 2.9 concerns a class of optimal circulant graphs GC(n; S=(sl, s2)) for which S

gives a graph with minimum diameter D for a given n. As discussed in Chapter 1, the

class G=GC(n;{sl =D, s2=D+l f ) with 2D2-2D+l<n5 2 ~ ~ + 2 ~ + 1 is one of the few known

optimal classes. In Lemma 2.2 we consider the case s2>_2-sl+l, sp-2, and then deal with

the remaining cases in Theorem 2.10.

Theorem 2.9

Let G=GC(n; {sl =D,s2=D+ I)), n12D2-2D+ 1. Then b' D>2, X (G)13.

Proof:

We use the proof of Theorem 2.8 with k=2.

If s 2 is even, then no =

If s 2 is odd, then no' = 1 .(3sk)=3.(D+ 1)12D2-2D 1, b' D21. Then 3-(2-s, - s, )

'd n> no', X ((353.

Remark 2.2

The bound 3.(D+l) in Theorem 2.9 is the best possible. If n=3(D+1)-1, then G contains a

clique of size 3 (from any vertex, follow a chord of length D and then two chords of

length D+1) and the order of G is not divisible by 3, so X (G)>3 by Theorem 2.6.

Lemma 2.2

For connection set S={sl, s21 s22.s1+1 and s1>2), 3 no such that 'dn> no, X (G=

GC(n;S))53.

Proof:

Our development of the colouring function C: V(G)+{a, b, c ) for G=GC(n;{sl,

s2)) consists of two cases: sl even and sl odd.

Case sl even:

We start with some simple restricted cases and build up a complete description of

the colouring function C. Let pattern P consist of 3 blocks: x consecutive vertices

coloured a, followed by x consecutive vertices coloured b, ending with x consecutive

S vertices coloured c, where 1 <xgl- l . Consider an s 2 such that there is a basic pattern Yb

2

4 consisting of repetitions of P, where s2'lYbl-2'~+z, 1 5 ~ 5 ~ - 1 , x=- . Since s2>2.sl+1, there 2

must be at least two repetitions of P in Yb. An example satisfying these conditions is

shown in Figure 16. In this example, sl=8, s2=17, x=4 and Yb contains two repetitions of

P.

Figure 16: Colouring scheme for GC(n=IYbl+(y,bl=49; (~1=8 ,s2=1 7)).

The construction of Yb guarantees that no two vertices at distance sl in Yb are coloured

by the same colour. (By distance we mean distance according to lexicographic ordering.)

Note that this property remains true if P is constructed of blocks of different lengths if the

S length of every block is between and sl-1. Let n= lYbl and note that s2=(s2-n) (mod n)

2

= Is2-nl, so the neighbours of any vertex u that are at distance s 2 in Yb are (u+ s;?) (mod n)

= (u+ s2-n) (mod n) and similarly for the neighbours at distance sl. Stated another way,

the graphs Gc(n;(sl, s2)) and Gc(n;(sl, Is2-nl)) are identical. Since any two blocks of the

same colour are separated by 2.x vertices of different colours and (x+l)<ls2-nl<(2x-1),

pattern Yb is a proper colouring of Gc(n=lYbl;(sl ,sz)). Moreover, Yb can be repeated to

get a proper colouring for Gc(n;{sl ,s2)) for any n=t.lYbl, t >1. This can be seen by

considering t=2. For any vertex u such that IYbl< u<2'lYb) there is a vertex v such that

O<v<lYbl and u=(v&IYbl) (mod n), hence C(v)=C(u). The neighbours of v at distance s 2 are

(v+s2) (mod n) = (v+(lybl-2*x+z)) (mod n)= (u &(-2.x+z)) (mod n) and these neighbours

are assigned different colours than u (and v) and similarly for the neighbours of v at

distance sl . Now, consider graph Gc(n=lYbl+l;{sl ,s2)) using the same sl and s 2 as above. An

extended pattern Yeb can be constructed from Yb by increasing the size of one block from

x to x+l. In the example of Figure 16, the first block of the first P has x+l consecutive

a's. In terns of yeb, s 2 = lyebl -~'x+z, O<Z~X-2. By an argument similar to the one above,

(x+2)51s2-n1<2x. Since, any two blocks of yeb of the same colour are separated by at least

2.x vertices of different colours, and any single block has length at most (x+l), pattern

yeb properly colours Gc(Yeb;{sl ,s2)). It is easy to verify that the same argument holds for

any modified pattern ymb obtained from Yb by increasing the size of any i>l blocks to

x+l as long as at least one block remains at length x, and the constraint s2'lYmbl-2.~+z,

O<z<x-2 is respected. Moreover, ymb can be repeated to get a proper colouring for

Gc(n;{sl ,s2)) for any n=t.IYmbl, t >l . Any pattern Ymb with i blocks of length x+l and the remaining j>l blocks of

length x can be extended to a pattern yemb (i.e. an extended modified pattern) with i+l

blocks of the length x+l and j - 1 blocks of length x. Using arguments similar to above,

Ymb followed by y e m b properly colours Gc(n=lymbl+l \Yembl;{sl,s2)). In the example at the

bottom of Figure 16, ymb on the right of the diagram has i = 0 and j = 6 and yemb on the

left has i =1 and j = 5. To generalize this construction, let no=(Iymbl-l)'lymbI and consider

any n>no, that can be written as n=t.lYmbl+r, with t>lYmbl-l and 05 r < lYmbl. Then r

repetitions of pattern Yemb followed by t - r repetitions of pattern Ymb gives a proper

colouring for GC(n; {sl, ~2)) .

The correctness of the colouring scheme presented so far depends on a constraint

on the relationship between Ymb and s2. What is left to be shown is that Vsl, s 2 3 y m b such

S 4 S that s 2 = lYmbl- ~.x+z, l<z<x-1, 1 5 x 5 ~ ~ - 1 . Let s2= q- - +r -1, O<r<1. Since s2>2-sl+l,

2 2 2

q>4. The construction of Ymb consists of three cases: q=0,1,2 (mod 3). If q=l (mod 3) and

r>2 then we can use Ymb'yb and x = L , and the constraint s2=lybl- 2x+z, I<z<x-~ is 2

respected. For all other cases, let s2= lYmbl- 2-x+z, llz5x-1 and set z=x-1.

Figure 17: Construction of Ymb from Yb for sl=8 and s2=28.

q=l (mod 3), r =0,1.

In the example shown in Figure 17, the chords sl=8 and s2=28 satisfy q=7, and

r=l. The depicted chords indicate two neighbours of vertex 0. The diagram on the

top shows pattern Yb, where each block is of the size 4. Pattern Yb is a proper

colouring of Gc(n=36; (8, 28)). If the size of any block in Yb is increased by 1,

then GC(n=37, (8, 28)) is not properly coloured. However, we obtain pattern Ymb

shown on the bottom of Figure 17 by uniformly increasing the size of each block

in Yb, until the neighbour of vertex 0 is the last vertex in the second last block.

This construction of Ymb allows the size of the last two blocks to be safely

increased by 1, yielding the construction of Yemb. Note that the number of blocks

between vertices 0 and 28 in the pattern Ymb has dropped by 2 relative to the

pattern Yb, while the number of blocks between vertices 0 and 8 remained

constant.

The chord length s 2 can be written as:

Term sl+r can be rewritten as (q-2). -- I;';] + rl for some O<rl<(q-2). Then,

s s l + r Hence, we use x=J + -

2 1,-21 , s2=(q-2).x +rl-1 and pattern Ymb consists of rl

blocks of length (x+l) and q-rl-1 blocks of length x.

S In the example in Figure 17, we start with each block of size '=4 (shown at the

2

s, + r 8+1 =1, as shown top of Figure 17). Then, we increase each block by - - -

1,-21-17-21

in the middle of Figure 17. Finally, we increase the first rl blocks by 1, rl =

s, + r (sl+r)- (q-2)1-] =(8+1)-5-1=4, as shown at the bottom of Figure 17.

q-2

q=O (mod 3).

Similarly, as in the previous case,

S l ir; (q-l)- + , where rl satisfies OYl<(q-1). Then, 2

length (x+l) and q-rl blocks of length x.

q=2 (mod 3).

As in the previous cases,

r= q- - + rl, where rl satisfies O<rl<q. Then, 1;J

S We use x='+ r , s2=qmx +rl-1 and pattern Ymb consists of rl blocks of length

2 LJ (x+l) and q-rl+l blocks of length x.

Case sl odd:

The proof for sl odd is similar to the case sl even, so we omit the details. The

main difference is that pattern P is defined in terms of two patterns Pi and P:! as in the

case s k odd in the proof of Theorem 2.8.

We have shown that Vsl, s23Ymb such that constraint s2= Iymbl-2'~+Z, l<z<x-1 is

satisfied. Therefore, 3 no such that V &no, GC(n ;(sl ,s2)) is 3-~0hrable . I

Theorem 2.10

For connection set S=(sl, s21 s2 >sl 21 and s2#2-sl), 3no such that VnL no,

X (G=GC(n;S)) < 3.

Proof

First, note that if s2=2-sl, then G has a clique of size 3, so G is not 3-colourable

for n=1,2 (mod 3) by Theorem 2.6. Now, suppose that s113 and consider two cases:

~212-sl-1 and s 2 L2.sl+l. The first case is covered by Theorem 2.8, and the second case is

covered by Lemma 2.2. Theorem 2.8 also covers the case sl=2, s2=3.

Now, consider the remaining cases sl=l, s2>3 and sl=2, s2.5. Let G'=GC(n';(j.sl,

j-s2)), ~ 2 3 . From the first part of this proof, 3 no' such that b' n'ao', X (G') 5 3. Choose

nb + X x such that no'+x =O (mod j) and 05 x < j and let no= - . For any nL no the graph

j

~ ' = G ~ o . ~ ; ( j . ~ ~ , j-s2)) consists of j copies of ~ '=G~(n; (s l , s2)) [32]. Since, j.n ?no',

X (G)= X (G') 5 3.

I

3 Efficient Domination in Circulant Graphs

In this chapter we study the existence of dominating sets and the relationship

between independent perfect dominating sets and a special case of graph homomorphism

commonly referred to as graph covering.

3.1 Introduction

In this section, we consider necessary conditions for Gc to admit efficient

domination. In Theorem 3.1 we show that an efficient dominating set is a minimum

dominating set if G admits efficient domination. The result of Theorem 3.2 implies that

the number of vertices in a z-regular graph must be a multiple of (z+l) if the graph admits

efficient domination. Finally, in Theorem 3.3 we give an alternative proof to the proof in

[28] that a circulant graph admits efficient domination if and only if it is a covering graph

of a complete graph.

Theorem 3.1

If G admits efficient domination, then y(G)=ID,I.

Proof:

Let D be a dominating set of G other then D,. D can be written as

D=(D\D,)U(DnD,). Suppose that D and D, are disjoint, DnDe=O. Since every vertex

outside of D, has a single neighbor in D,, it follows that the vertices of D, have no

common neighbors in D. Therefore, no vertex in D can dominate more than one vertex in

D, and IDI?IDel. Similarly, if DnDe#O, then no vertex in DnD, can dominate any vertex

in DeD. Hence, ID\D,I?ID,\DI and it follows that IDI?ID,(. Therefore, if D, is a

dominating set in G, then y(G)=ID,(. I

Definition 3.1

A graph G is said to cover a graph H if there is a function (called a covering function) g:

V(G)+V(H) which preserves the neighborhood of any vertex v of G, (g(u)lu~N~(v)} =

Note that graph covering is a restricted version of graph homomorphism. In the

definition of graph homomorphism given in Chapter 1, the only restriction placed on the

mapping function h is that the labeling of vertices of G preserves the structure of H. The

definition of graph covering adds two additional properties to graph homomorphism;

informally:

A vertex in G, and its corresponding vertex in H with respect to the mapping

function h, have the same degree.

For every vertex v in G, the mapping function h assigns a different label to each

of its neighbors.

Example 3.1

Figure 18: Hypercube Qj is a covering of the complete graph Kq.

Let H be the complete graph on 4 vertices and G be the hypercube on 8 vertices. The left

of Figure 18 portrays graph H with vertex set V(H)={red, green, blue, yellow). The

mapping V(G)+V(H) showing that G is a covering of H is presented on the right of

Figure 1 8.

It is important to point out that in the example shown in Figure 18, graph H is a

complete graph. In fact, the reminder of this chapter deals with the efficient domination

problem in circulant graphs which, according to Theorem 3.3 below, is equivalent to the

graph covering problem when H is a complete graph.

Theorem 3.2

Let G=GC(n;S) be a z-regular graph. If G admits efficient domination, then n=O mod

(z+l).

Proof:

Let D, cV(G), D, = {do, dl, d2, ..., d(i-1)) be an efficient dominating set of

G=Gc(n;S). Every vertex in G is either a dominator or a vertex dominated by a member of

a dominating set D,. Hence, V(G)=DeUN(do)UN(dl)U---UN(d( i-l)). Since D, is an

independent and perfect dominating set of G, it follows that Vd,, 4 E D,, if t f j then

N(d,)flN(dj)=W. Hence, IV(G)I=i+i.(z)=i.(z+l).

I

Theorem 3.3

Let G=Gc(n=i(z+l); S), i~ z', S={sl, s2,. . ., sk), be a z-regular graph. Graph G admits

efficient domination iff G is a covering of the complete graph

Proof:

=+

Let G admit efficient domination, and let D,(O) cV(G), D,(O) ={do, dl, d2,. . .,

d+l)) be an efficient dominating set of G.

Consider vertices u'=u+r and v'=v+r, such that u , v ~ D,, 1lrFn-1. Vertices u' and

v' do not share an edge because lu'-v'l=l(u+r)-(v+r)I=Iu-VI @S. Since u and v have no

common neighbors (i.e. b' s,, sy E S, u+s.#v+sy), it follows that vertices u' and v' have no

common neighbors either (i.e. b's,, sye S, (u+r)+s&(v+r)+sy). In other words, N(do+r),

i

N(dl+r), N(d2+r), . . . , N(d(,l)+r) are mutually disjoint and I N(dj + r) 1 =n - i. Therefore j = O

De(r)={do+r, dl+r, d2+r,.. ., d(,-l)+r), is an efficient dominating set of G for any l<rLn-1.

Note that the same result can be observed by shifting a known dominating set

clockwise/counterclockwise by an arbitrary constant in the graphical representation of G,

because the structure of the graph G is independent of its labeling. Let the vertices of G

be partitioned into the following (z+l) efficient dominating sets:

Without loss of generality, consider vertex u E De(sl), and vertices v, w E N(u).

Suppose that the suggested partitioning of V(G) places both v and w into the same set,

e.g. De(sj), sj#sl. Since De(sj) is also a dominating set of G, then the suggested

arrangement implies that vertex u is dominated by both v and w, contradicting the

efficiency property of De(sj). Moreover, by definition, if veN(u) then vBDe(sl).

Therefore, the defined partitioning of V(G) along with IN(u)l= deg(u)=z indicates that u

has exactly one neighbor in each partition, other than its own.

Without loss of generality, let the vertices of Kz+l be labeled from 0 to z and let

u E De(a) and v E De(b), a, b E (0, sl,. . . , sk, -sl,. . . , -sk). Also, let E,:V(G)+ {O,l,. . . z) be a

mapping function such that E,(u)= E(v) iff a=b (function E, simply assigns a different label

to each of the partitions). Clearly b'u, v, WEV(G), if v, WEN@) then E,(v)#t(w) because u

has exactly one neighbor in each partition other than its own. In addition, the degree of

every vertex in G is z and edge (E,(u), E,(v)) E Kz+l. We conclude that G is a covering of a

complete graph Kdl.

Let the covering function g partition the vertices of G into sets Lo, L1, ..., L,.

Clearly, since G covers Kdl then b ' v ~ Li, if if0 then3 !u, u E N(v) such that u E Lo, Hence,

Lo is a perfect dominating set of G. Moreover, VVELO, if u ~ N ( v ) then u$Lo and the

vertices in the partition Lo form an independent set in G. Therefore Lo is an independent

perfect dominating set. I

3.2 Circulant Graphs with Two Chord Lengths

In this section we completely characterize which circulant graphs with two chord

lengths have an efficient dominating set D,. Our approach to the question of domination

relies on the highly regular structure of circulant graphs. The technique that leads to the

complete characterization is based on the periodic behavior of dominating sets. The

following proofs give sufficient and necessary conditions for the existence of a

dominating set as well as its exact structure (i.e. period) according to the relationship

between the two chord lengths. In particular, Lemma 3.1 deals with 4-regular circulant

graphs and the case when at least one of the chord lengths is relatively prime with the

number of vertices. On the other hand, Lemma 3.2 covers the case when none of the

chord lengths is relatively prime with the number of vertices and Gc is a 4-regular graph.

In Theorem 3.4 we give sufficient and necessary conditions for 4-regular GC(n;S) to

admit efficient domination. Finally, in Theorem 3.5 we provide a complete

characterization of 3-regular circulant graphs which admit an efficient dominating set.

Lemma 3.1

Let G=GC(n=5.i; S={sl,sz)), i~ Z+ be a connected 4-regular graph such that gcd(sl,n)=l

andlor gcd(s2,n)=l. If G admits efficient domination then Isl*s2l#O (mod 5) and sl,s2#O

(mod 5).

Proof:

Let D, ={do, dl, d2, ..., d(i-1)) be an efficient dominating set of G. Without loss of

generality let gcd(sl,n)= 1. Then all products assl, a=O, 1,. . .n-1 belong to different classes

(mod n). Therefore G has a Hamiltonian cycle: C=(O, sl, 2.sl,.. ., (n-1)-sl).

Suppose that the members of dominating set D, are not equally spaced along

Hamiltonian cycle C. With respect to C there are 4 cases to consider:

Case 1: 3 dx, d, E D, such that dx is adjacent to d, in C (i.e. dY=dx+sl)

This arrangement of dominators dx and d, contradicts the independence

property of D,.

Case 2: 3 dx, d, E De such that dx and d, are at distance 2 in C (i.e. d,=dx+2.sl).

Then 3 u E C, u=dx+ sl such that u is dominated by both dx and d,,

violating the efficiency property of D,.

Case 3. 3 dx7 d, E De such that dx and d, are at distance 3 in C (i.e. dy=dx+3-sl).

Let P,=uo, ul, 242, u3 be the path of length 3 in Hamiltonian cycle C such

that dx=uo7 dy=u3. Also, let v ~ = u ~ + s ~ ~ ~ j = ~ j + 2 ' ~ ~ for 0 1 i 53 (see Figure 19). Figure

19 could be showing some vertices of G more then once depending on the

relationship between sl and s 2 (i.e. if sl=2.s2 then ul and wo represent the same

vertex).

outlined gray (vertical) lines represent chord of length sl

solid black (horizontal) lines represent chord of length s:,

Vertices colored gray (white) are members (not members) of dominating set D.

Figure 19: Unobtainable construction of De with 2 members at distance 3.

Consider N[vl]=(vl, ul, vo, wl, 172). Vertex vl is neither dominated by dx or

d,, nor can it be a member of dominating set D, because vertex vo€N(dX)n N(vl).

Both ul and vo are dominated by dx and therefore cannot be members of D e .

Moreover, v2 cannot be in De because vertex v3 is a common neighbor of d, and

v2. Some member of De must dominate vl, so wl must be a member of D, .

Analogously, vertex v2 can be dominated only by vertex w2. However the

adjacency of wl and w2 contradicts the efficiency property of De.

Case 4: 3 dx, d, E De such that dx and d, are at distance 4 in C (i.e. d,=dx+4.sl).

Let Pu=uo, ul, 242, ug, u4 be the path of length 4 in Harniltonian cycle C

such that dX=uo, dy=u4. Also, let v ~ u ~ + s ~ , w ~ = u ~ + ~ ' s ~ and zz=ui+3-s2, O< i 14 (see

Figure 20). Please note that Figure 20 could be showing some vertices of G more

than once, depending on the relationship between sl and s2.

Vertex u2=dx+2-s1 must be dominated either through the chord to u2+s2 or

through the chord to u2-s2. Since the two cases involve equivalent arguments,

assume that u2 is dominated by v2=~2+s2. If v2=dx or v2=dy then it must be the case

that w4=dx or wO=dy which according to Case 2 contradicts the efficiency property

of De. Therefore v2 E D e , and v2#dX,d,.

Consider N[wl]=(wl, vl, wo, zl, w2). Vertex wl cannot be a member of De

because v1€N(v2). Both vl and wz are dominated by v2 and therefore cannot be

members of D,. Moreover, wo cannot dominate wl because vo is the common

neighbor of d, and wo. Therefore vertex zl must be a member of De. Similarly,

vertex w3 can be dominated only by 23. However, vertices zl and 23 are at distance

2 in C which according to Case 2 contradicts the efficiency property of D e .

outlined gray (vertical) lines represent chord of length sl

solid black (horizontal) lines represent chord of length s2

Vertices colored gray (white) are members (not members) of dominating set D

Figure 20: Unobtainable construction of De with 2 members at distance 4.

The four cases exhaust all possibilities for unequally spaced dominators yielding

the necessity of equally spaced dominators with respect to Hamiltonian cycle C.

Since n is a multiple of 5 and gcd(sl,n)=l , it follows that gcd(sl,5)=l. Consider

cycle C with respect to the (mod 5) operation: 0 (mod 5), sl (mod 5), 2 .~1 (mod 5),. . ., (n-

1 ) ~ s ~ (mod 5). The vertices with indices belonging to the same congruence class are

equally spaced along C (i.e. every fifth vertex along C is 0 (mod 5)).

Since vertices in D, are also equally spaced along C, then D, ={0,5,. .., 5(i-1)) is

one such set. Moreover, since each vertex ue D, is uniquely dominated, there does not

exist a path P=vj, u, vk in G such that vj, V ~ E De. In other words, given that vertices at

distance 2 cannot belong to the same partition (i.e. same congruence class) it follows that

Is1ks2(#0 (mod 5). Moreover, the independence property of D, implies that sl,s2#0 (mod

5). I

Before proceeding with further discussion, it is beneficial to consider an

alternative graphical representation of a connected graph Gc(n;S={sl,s2)), where

gcd(sl,n)#l or gcd(s2,n)#l, as first suggested in [6].

Observation 3.1

Let G=GC(n; {sl, s2)) and suppose that gcd(sl,n)= a, a>l. Then n= a$, for p = z . a

We partition the vertices of G into the following sets:

All the vertices within a single set are in different congruence classes (mod n)

because n=a- p. Since sl is a multiple of a then vertices belonging to the same set are in

the same congruence class mod a. It follows that all the sets are mutually disjoint.

In fact, each of the sets represents a cycle in G. The cycles are obtained by

starting at an arbitrary vertex v, and traversing the path consisting of chords of length sl

until vertex v is reached again. We denote such cycles with Csl(b), O<b<a-1. In addition,

the cycles Csl(b) have order P. For example, graph GC(42; (6, 11)) shown in Figure 21

has a=6 and beta P=7. Vertex 0 belongs to the vertical cycle 0, 6, 12, 24, 30, 36

consisting only of chords of length 6. In the proposed graphical representation of G, we

label vertical cycles from left to right with Csl(0), Csl(l),. . ., Csl(a-1), where the leftmost

cycle Csl(0) contains the vertex labeled 0 (see Figure 21). Similarly we can obtain cycles

constructed by traversing only the chords of length sz.

outlined gray (vertical) lines represent chord of length s,

solid black (horizontal) lines represent chord of length s2

Figure 2 1 : Graphical representation of G=Gc(42;

Choosing an arbitrary vertex v and traversing a chords of length s 2 ends at the

vertex u=v+ a -s;! (mod n). Since vertices u and v are in the same congruence class mod a,

then u and v belong to the same cycle Csl(b), for some O<b<a-1. Therefore, 3u and a p

with 0*< p such that u=v+p-sl (mod n). Hence, v+ a *s2 (mod n)= v+p-sl (mod n) which

implies a .s2 (mod n)= p-sl (mod n). We use ,us to denote the least nonnegative residue of

P (mod PI. It is interesting to point out that G=Gc(n; (sl, s2)) with ,us equal to zero is

isomorphic to the 2D torus, Tx8, with dimensions x= a and y= P. The graph shown in

Figure 21 can be regarded as a skewed (shifted) 2D torus [22] with skew ps=4. I

Lemma 3.2

Let G=GC(n=5i; S=(slg2)), ~ E Z ' be a connected graph such that gcd(sl,n)#l and

gcd(s2,n)#l. If G admits efficient domination then Isl+s21#0 (mod 5) and sl,s2#0 (mod 5).

Proof:

Let n=a$ and gcd(sl,n)=a, a>l . From the connectivity property of G, gcd(sl, s2,

n)=l, it follows that sl and s 2 cannot both be multiples of 5 (i.e. gcd(sl,5)=l or

gcd(s2,5)=l). Without loss of generality let gcd(sl,5)=1. Since sl is a multiple of a then

gcd(a,5)=l. Therefore, P must be a multiple of 5, P=j.5 for some 121.

Suppose that there exists a cycle Csl(b), for some Oib3-1, such that the number

of dominators contained in Csl(b) is greater then j. Then 3dx, dy E D,, dx, dy E Csl(b)

such that dx and dy are at distance less than 5. Consider path P in Csl(b) starting at vertex

dx and ending at vertex dy such that P has length less then 5. The same argument as the

one presented in Lemma 3.1 can be used to prove that dx and dy violate the efficiency

property of D,. Therefore, the number of dominators in Csl(b), O<b<a-1 is exactly j.

Moreover, the same argument implies that any two members of a dominating set

contained in Csl(b) are at distance 5.2 from each other, for some z=1, 2, . .. j-1. In other

words, every path of length 4 in Csl(b) contains exactly one member of the dominating

set D,.

The value of a (i.e. a>l) guarantees the existence of at least 2 vertical cycles, i.e.

Csl(0) and Csl(l). Consider vertex dX€Dc, belonging to the cycle Csl(0) depicted on the

left of Figure 22. Since dx dominates dx+s2, vertices dx+s2+sl cannot be members of De

because dx+s2 is a common neighbor of d, and dx+s2*sl. Given that every path of length 4

contains a single dominator, either u=dx+s2 -2-sl or v=dx+s2 +2.sl must belong to the set

De. Without loss of generality let VED,.

outlined gray (vertical) lines represent chord of length s,

solid black (horizontal) lines represent chord of length s2

Vertices colored gray (white) are members (not members) of dominating set De.

Figure 22: Periodic behavior pattern of the members of De.

By the same argument either vertex u'=v+s2-2.sl=dx+2.s2 or v'=v+s2

+2-sl=dx+2s2+4-s1 (see center diagram in Figure 22) must belong to De. Since u' and dx

have a neighbor in common, namely vertex dx+~2, it follows that vertex v' must belong to

De. (Note: in the case that a=2, cycle Csl(2) can be viewed as a duplicated cycle Csl(0).)

In general, the argument can be extended for any value of a and this shows the

emergence of the placement pattern (i.e. s2+2.sl) of the dominators in the graphical

representation of G. Furthermore, if @6, vertex v"=dx+5.sz+10.sl E Cs1(5) must also

belong to D,. This implies that vertex u"= dx+5-s2 is a member of D, because the

members of D, on a single cycle Csl(b), Osbla-1 are at distance 5.2, z=1, 2, . . . j-1. As a

result, the members of D, exhibit a repetitive pattern with respect to their location in the

horizontal cycles of G withperiod of repetition equal to 5.

The choice UED,, shown on the right of the Figure 22, leads to another pattern that is a

mirror image of the pattern just discussed. It can be obtained by replacing chord length s2

with -s2 (i.e. -s2 +2-sl).

Therefore, to analyze the behavior of D,, it is sufficient to restrict our attention to the

5 cases which correspond to the values of a (mod 5).

Case: a = 4 (mod 5)

To simplifl the following argument (i.e. avoid illustration of wraparound edges),

Figure 23 depicts cycle Csl(0) twice. Note that the location of the dominators in

cycle Cs1(3) is equivalent to the location of dominators in cycle Csl(a-1) because

3 and (a-1) belong to the same congruence class (mod 5).

Consider path Po=do, do+sl, do+2.sl, do+3.Sl, do+4.Sl consisting of 5

consecutive vertices in Csl(0). Vertices do+sl and do+4-sl are dominated by do and

do+5-sl respectively while vertex do+2.sl is dominated by vertex dleCsl(l). The

only vertex fiom Po not yet considered is vertex do+3.Sl. Since the three neighbors

of do+3.sl: vertices do+2-sl, do+4.Sl and d0+3-s1+s2 are all dominated, there must

be a vertex located in cycle Csl(a-1) which dominates do+3.sl. Figure 23 shows

dominator daml to be the one dominating vertex do+3.sl, yielding p, =2 (recall that

,us is the skew in the graphical representation of G) . However, the existence of the

periodic behavior noted earlier implies that pd2+5.c also respects the properties

of dominating set De, for any O<c5j-1.

outlined gray (vertical) lines represent chord of length s,

solid black (horizontal) lines represent chord of length sz

Vertices colored gray (white) are members (not members) of dominating set D.

Figure 23: Graphical representation of G for a = 4 (mod 5).

As stated in Observation 3.1, p, must satisfy a as2 (mod n)= p.sl (mod n)

which implies that a .s2 (mod 5)= p-sl (mod 5). Substituting p,= 2 + 5 ~ into this

equation gives:

(2+5.c)-sl(mod 5) = 4.s2(mod 5 ) ~ 2.sl(mod 5)=4.s2(mod 5) H

sl(mod 5)= 2-s2(mod 5).

The second (i.e. mirror image) pattern leads to the congruence:

(2+5.c).sl(mod 5)= 4.(-s2) (mod 5) w 231 (mod 5)=-4.sz(mod 5) e

sl(mod 5)=-2.s2 (mod 5) w sl(mod 5) = 3.~2 (mod 5).

outlined gray (vertical) lines represent chord of length sl

solid black (horizontal) lines represent chord of length sz

Vertices colored gray (white) are members (not members) of dominating set D.

Figure 24: Graphical representation of G for a = 3 (mod 5).

Case: a = 3 (mod 5)

The pattern shown in Figure 24 corresponds to the skew ,us= (-1) (mod 5) = 4

(mod 5). Therefore, the following congruence is obtained:

4-sl(mod 5)= 3.s2 (mod 5) w sl(mod 5)= 2.s2(mod 5).

The second pattern leads to the congruence:

4-sl(mod 5)= 3. (-sz) (mod 5) w 4.sl(mod 5)=-3-s2(mod 5) H

Case: a= 2 (mod 5)

The pattern shown in Figure 25 corresponds to the skew pS=l (mod 5).

outlined gray (vertical) lines represent chord of length sl

solid black (horizontal) lines represent chord of length s2

Vertices colored gray (white) are members (not members) of dominating set D

Figure 25: Graphical representation of G for a = 2 (mod 5).

Therefore, the following congruence is obtained:

sl(mod 5)= 2-s2(mod 5).

The second pattern leads to the congruence:

sl(mod 5)= 2.(-s2) (mod 5) H sl(mod 5)=-2.s2(mod 5) H

Case: a = 1 (mod 5)

Note: Since a>l it must be the case that a26.

outlined gray (vertical) lines represent chord of length sl

solid black (horizontal) lines represent chord of length s2

Vertices colored gray (white) are members (not members) of dominating set D.

Figure 26: Graphical representation of G for a = 1 (mod 5).

The pattern shown in Figure 26 corresponds to the skew p,=3 (mod 5).

Therefore, the following congruence is obtained:

3.sl(mod 5) = s2(mod 5) @ sl(mod 5)= 2.s2(mod 5).

The second pattern leads to the congruence:

3.sl(mod 5)= 1-(-s2) (mod 5) @ 3.sl(mod 5)=-s2(mod 5) @

Case: a = 0 (mod 5)

Contradicts gcd(sl,5)=l

In conclusion, regardless of the value of a, the congruences sl(mod 5)= 2-s2(mod

5) and sl(mod 5) = 3-s2(mod 5) are obtained. The following sets are obtained:

The pair (sl(mod 5), s 2 (mod 5)) = (0, 0) satisfies both congruences but contradicts the

condition gcd(sl,5)=1, and hence is not a valid solution. Therefore, if G admits efficient

domination, then the valid solution pairs (sl,s2) from sets A and B give Islfs21#0 (mod 5)

and SI ,s2#O (mod 5).

I

Theorem 3.4

Let G=GC(n=5i; {sl, s2)), i~ Z+ be a connected 4-regular graph. G admits efficient

domination iff Is1+s21#0 (mod 5) and sl,s2#0 (mod 5).

Proof:

=>

Follows from Lemma 3.1 and Lemma 3.2.

<=

Let A={O, 5, 10.. ., 5(i-I)), and u E V(G)M. Since sl,s2#0 (mod 5), then A forms

an independent set in G. Vertex u can be written as u=5+q+r, 0 3 4 - 1 and l<r<4. From

Isl*s2l#O (mod 5) it follows that the chord lengths sl, s2, -sl, and -s2 are in 4 different

congruence classes (mod 5) (other than class 0 (mod 5) because s1,s2#0 (mod 5)).

Therefore, exactly one chord length is in the same congruence class (mod 5) as -r, and u

has a single neighbor in A. Hence A is an efficient dominating set of G. I

Example 3.4

Let G=GC(20; (3, 6)). According to Theorem 3.4, G admits efficient domination

and it is a covering of the complete graph Kg by Theorem 3.3. The covering function

g:V(G)+ {0,1,2,3,4), g(v)=v (mod 5) is shown in Figure 27.

Figure 27: G=GC(20; {3,6)) as a covering of Kg.

The following result is proved [21]. It also follows from the proof of Lemma 3.2.

Corollary 3.1

Let x 23 and y23. Then the torus T,, admits efficient domination iff x=5.i, y=5-j, i, j EZ'.

Proof:

A torus is a Cartesian product of two cycles and in the graphical representation, a

2D torus T,, has skew ,us=O. The proof of Lemma 3.2 can be used to establish necessary

and sufficient conditions for TXJy to admit efficient domination. (=) Skew ps=O implies

that the number of vertical cycles Csl(b), Osblx-1 has to be a multiple of 5. Also,

ICsl(b)l=y must be a multiple of 5 itself. ( c ) If x=5-i, y=57 then either of the two

patterns specified in Lemma 3.2 provides efficient domination of T,, because the

patterns are proved to have period of repetition equal to 5 . I

Theorem 3.5

Let G=GC(n=4i; {sl, sz=")), i~ Z' be a connected 3-regular graph. G admits efficient 2

domination iff n= 8j +4, j~ 2'.

Proof:

that sl has to be an odd integer relatively prime with n.

Let D, ={do, dl, d2,. . ., d(i-l)) be an efficient dominating set of G. All products a-sl,

a=O, 1,. . . n- 1 belong to different classes (mod n) and G has a Hamiltonian cycle: C=(O, s 1,

2-sl,.. ., (n-l).sl). A similar argument to the one presented in Lemma 3.1 can be used to

prove that the members of dominating set D, have to be equally spaced along

Hamiltonian cycle C. Consider cycle C with respect to the (mod 4) operation: 0 (mod 4),

sl (mod 4), 2.sl (mod 4), ..., (n-l).sl (mod 4). Then vertices that belong to the same

congruence class are equally spaced along C (i.e. every fourth vertex along C is 0 (mod

4)). Since vertices in D, are also equally spaced along C, then D, ={0, 4, ..., 4.(i-1)) is

one such set. The independence property of D, implies that s2= cannot be a multiple of 2

4. Since n is a multiple of 4 then it must be the case that n= 8j +4 for some j~ z'.

Suppose that n= 8j +4 for some j~ Z+ and let A={O, 4, 8.. . , 89) and u E V(G)\A. Since sl

is an odd integer and s2= 4j +2 then A forms an independent set in G. As in Theorem 3.4,

vertex u can be written as u=4*q+r, O5q5i-1 and 1953. Since s2=2 (mod 4) and sl is odd,

it follows that the chord lengths sl, s2, -sl, are in 3 different congruence classes (mod 4)

other than class 0 (mod 4). Therefore, exactly one chord length is in the same congruence

class (mod 4) as -r, and u has a single neighbor in A. Hence A is an efficient dominating

set of G. I

4 Conclusions and Open Problems

In this thesis, we have established the existence of infinite families of 3-colourable

circulant graphs. In particular, we proved that X (GC(n; S))F 3 for all sufficiently large n

whenever S = ( S ~ , . . .skJ s,t 2sk-1 ... >st11 and st? 31 and skf2sl } or S=(sl,s21 s 2 >

>slL1 and s2#2s1 ). We also established that X (G) 5 3 for recursive circulant graphs

RGC(n; d), n = c.dm , 1 < c l d and for a class of optimal double loop graphs. We also

provided necessary and sufficient conditions for 3-regular and 4-regular circulant graphs

GC(n; {s1,s2}), to admit an efficient dominating set.

Some interesting problems concerning the chromatic number and efficient

domination of circulant graphs that remain open are:

Find sufficient and necessary conditions for circulant graphs GC(n;S), (S)>2 to

admit efficient domination.

Find the minimum no such that Vn2no, GC(n; isl, s2)) has minimum

chromaticity.

Given a cardinality k, find sets S with IS1 = k such that X (GC(n; S))< 3.

For an arbitrary set S, determine if there exists a non-trivial value x such that

\d n>_no, X (GC(n; S)) 5x.

References

[I] B. Alspach, "Problems", Discrete Mathematics, 50: 115, 1984.

[2] K. Appel, W. Haken, "Every planar map is four colorable. Part I:

Discharging", Illinois Journal of Mathematics, 2 l(3): 429-490, 1977.

[3] K. Appel, W. Haken, J. Koch, "Every planar map is four colorable. Part 11:

Reducibility", Illinois Journal of Mathematics, 2 l(3): 491 -567, 1977.

[4] F. Baccelli, G. Cohen, G. Olsder, J. Quadrat, Synchronization and Linearity:

An Algebra for Discrete Event Systems, John Wiley and Sons, England, 1992.

[5] J.-C. Bermond, F. Comellas, D.F. Hsu, "Distributed loop computer networks:

A survey", Journal of Parallel and Distributed Computing, 24:2-10, 1995.

[6] J.-C. Bermond, 0. Favaron, M. MahCo, "Hamiltonian decomposition of

Cayley graphs of degree 4", Journal of Combinatorial Theory, Series B, 46: 142-

153,1989.

[7] J.-C. Bermond, G. Illiades, C. Peyrat, "An optimization problem in distributed

loop computer networks", Proc. 3rd International Conference on Combinatorial

Mathematics, New York, June 1985, Annals of the New York Academy of

Sciences, 555:45-55, 1989.

[8] D.K. Biss, "Hamiltonian decomposition of recursive circulant graphs",

Discrete Mathematics, 2 14: 89-99,2000.

[9] J.-Y. Cai, G. Havas, B. Mans, A. Nerurkar, J.-P. Seifert, I. Shparlinski, "On

routing in circulant graphs", Proc. 5th International Computing and

Combinatorics Conference (COCOON'99), Lecture Notes in Computer Science,

vol. 1627, T. Asano, H. Imai, D.T. Lee, S. Nakano, T. Tokuyama (Eds.),

Springer-Verlag, 1999,3 60-369.

[lo] C. Chou, D.J. Gum, K. Wang, "A dynamic fault-tolerant message routing

algorithm for double-loop networks", Information Processing Letters 70: 259-

264,1999.

[l 11 B. Codenotti, I. Gerace, S. Vigna, "Hardness results and spectral techniques

for combinatorial problems on circulant graphs", Linear Algebra Applications,

285(1-3):123-142, 1998.

[12] T. H. Cormen, C. E. Leiserson, R. L. Rivest, Introduction to Algorithms,

MIT Press, Cambridge, MA, 1990.

[13] H. Debrunner, H. Hadwiger, V. Klee, "Combinatorial Geometry in the

Plane", Holt Rinehart and Winston, New York, 1964.

[14] I.J. Dejter, 0. Serra, "Efficient dominating sets in Cayley graphs", Discrete

Applied Mathematics, 129(2-3): 3 19-328,2003.

[15] W. Deuber, X. Zhu, "The chromatic number of distance graphs", Discrete

Mathematics, 1651166: 195-204, 1997.

[I61 R. B. Eggleton, P. Erdos, D. K. Skilton, "Coloring prime distance graphs",

Graphs and Combinatorics, 6: 17-32, 1990.

[17] F. GiSbel , E.A. Neutel, "Cyclic graphs", Discrete Applied Mathematics,

99:3-12,2000.

[18] M. R. Garey, D. S. Johnson, H. C. So, "An application of graph coloring to

printed circuit testing", IEEE Transactions on Circuits and Systems, CAS-

23:591-599, 1976.

[19] C. Godsil, G. Royle, Algebraic Graph Theory: Graduate Texts in

Mathematics 207, Springer, New York, 2001.

[20] M. T. Goodrich, "Efficient and secure network routing algorithms."

Provisional patent filing, January 200 1.

[21] W. Gu, X. Jia, J. Shen, "Independent Perfect Domination Sets in Meshes,

Tori and Trees", preprint 2002,

available at http://erdos.math.swt.edu/research/research.htm

[22] P. Gvozdjak, J.G. Peters, "Modelling links in inclined LEO satellite

networks", SIROCCO 8, July 2001, Val de Nuria, 195-208.

[23] T.W. Haynes, S.T. Hedetnimi, P.J. Slater, Fundamentals of Domination in

Graphs, Marcel Dekker, Inc, New York, 1998.

[24] M.-C. Heydemann, "Cayley graphs and interconnection networks" in Graph

Symmetry: Algebraic Methods and Applications, Kluwer Academic Publishers,

Dordrecht, 1997.

[25] T.R. Jensen, B. Toft, Graph Coloring Problems, John Wiley & Sons, New

York, 1995.

[26] ] A. Kernnitz, H. Kolberg, "Coloring of integer distance graphs", Discrete

Mathematics,l91: 1 13-123, 1998.

[27] J. Kratochvil, A. Proskurowski, J. A. Telle, "On the complexity of graph

covering problems", Nordic Journal of Computing, 5(3): 173- 195, 1998.

[28] J. Lee, "Independent perfect domination sets in Cayley graphs", Journal of

Graph Theory, 37(4):213-219,2001.

[29] A. L. Liestman, J. Opatrny, M. Zaragoza, "Network properties of double and

triple fixed-step graphs", International Journal of Foundations of Computer

Science, 9:57-76,1998.

[30] M. Livingston, Q.F. Stout, "Constant time computation of minimum

dominating sets", Congressus Numerantium, 105: 1 16- 128, 1994.

[3 11 M. Livingston, Q.F. Stout, "Perfect dominating sets" Congressus

Numerantiurn,. 79: 187-203, 1990.

[32] B. Mans, F. Pappalardi, I. Shparlinski, "On the spectral Adam property for

circulant graphs", Discrete Mathematics, 254:309-329,2002.

1331 J.D. Masters, Q.F. Stout, D.M. Van Wieren, "Unique domination in cross-

product graphs", Congressus Numerantium, 1 1 8: 49-7 1,1996.

[34] P.K. McKinley, Y. Tsai, "A dominating set model for broadcast in all-port

wormhole-routed 2D mesh networks", International Conference on

Supercomputing 1994: 126- 1 3 5.

[35] L. Moser, W. Moser, "Solution to problem lo", Canadian Mathematical

Bulletin, 4: 187-1 89, 196 1.

[36] M.E. Muzychuk, G. Tinhofer, "Recognizing circulant graphs in polynomial

time: An application of association schemes", Electronic Journal of

Combinatorics, 8 (2001), #R26.

[37] J.-H. Park, K.-Y. Chwa, "Recursive circulant: A new topology for

multicomputer networks", Proc. IEEE International Symposium on Parallel

Architectures, Algorithms and Networks (ISPAN'94), Japan, 73-80, 1994.

[38] B. Robic, "Optimal routing in 2-jump circulant networks", Tech. Rep.

UCAM-TR 403-1996-06, University of Cambridge Computer Laboratory, Jun

1996.

[39] A. Spalding, "Min-Plus algebra and graph domination", PhD. Thesis,

University of Colorado, Denver, 1998.

[40] M. Trick, "Network resources for coloring a graph: A brief survey of

applications and algorithms",

available at http://mat.gsia.cmu.edu/COLOR~color.html

[41] J. Xu, Theory and Application of Graphs, Kluwer Academic Publishers,

Dordrecht, 2003.

[42] J. Xu, Topological Structure and Analysis of Interconnection Networks,

Kluwer Academic Publishers, Dordrecht, 2001.

[43] D.B. West, Introduction to Graph Theory, 2nd ed. Prentice Hall, New Jersey,

2001.

[44] X. Zhu, "Colouring of distance graphs", manuscript, 1995.