Chapter 1 Chapter 1 Fundamental Concepts IIFundamental Concepts IIPao-Lien Lai

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DefinitionsCountingThe pigeonhole principleGraphic sequencesDegrees and digraphs

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DefinitionsDefinitions

degree of v : ◦number of non-loop edges containing v plus twice the number

of loops containing v.

(G) : (\Delta) maximum degree of G.(G) : (\delta) minimum degree of G.k-regular : (G) = (G) = k .

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DefinitionsDefinitions

Isolated vertex : degree=0.Neighborhood : NG(v) , NG[v]n(G), |G| :

◦order of G , is the number of vertices in G.e(G) : the number of edges in G.

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CountingCounting5

(Degree Sum Formula) If G is a graph with vertex degree d1,…,dn,

then the summation of all di = 2e(G).

)()(2)(

GVvGevd

CountingCounting

In a graph G, the average vertex degree is , and hence

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)(

)(2

Gn

Ge

)()(

)(2)( G

Gn

GeG

Every graph has an even number of vertices of odd degree.

No graph of odd order is regular with odd degree.

A k-regular graph with n vertices has nk/2 edges.

Example Example

k-dimensional cube (hypercube Qk)Vertices: k-tuples with entries in {0,1} Edges: the pairs of k-tuples that differ in exactly one position. j-dimensional subcube: a subgraph isomorphic to Qj.

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Q3

ExampleExample

Structure of hypercubes◦Parity of vertex: the number of 1s◦Two independent sets

Each edge of Qk has an even vertex and an odd vertex. Bipartite graph

◦k-regular◦n(Qk)=2k. e(Qk)=k2k-1.

◦Two subgraphs of Q3 are isomorphic to Q2.

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The Graph MenagerieThe Graph Menagerie 動物園動物園

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triangle claw 爪 paw 爪子 kite 鳶

Petersen graphPetersen graph

The simple graph whoseVertices:

◦2-element subsets of 5-element setEdges :

◦the pairs of disjoint 2-element subsets

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12

3445

23 51

3552

2441

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The pigeonhole principleThe pigeonhole principle13

(Pigeonhole Principle) If a set consisting of more than kn objects is partitioned into n classes, then some class receives more than k objects.

Theorem1:Every simple graph with at least two vertices has two vertices of equal degree.

{0,1,……,n-1} 0 and n-1 both occurs impossibly

The pigeonhole principleThe pigeonhole principle14

Theorem 2:If G is a simple graph of n vertices with (G) (n-1)/2, then G is connected.

Example Example

Let G be the n-vertex graph with components isomorphic to and .

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2/nK 2/nK

2/nK 2/nK

12/)( nG

G is disconnected

* Induction trap* Induction trap

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Every 3-regular simple connected graph has no cut-edge.

False conclusion!!

CounterexampleCut edge

Degree sequenceDegree sequence17

degree sequence : the list of vertex degrees, in nonincreasing order, d1…dn.

Proposition Proposition

The nonnegative integers d1, d2, …, dn are the vertex degrees of some graph if and only if is even.

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n

i id1

Graphic sequencesGraphic sequences19

graphic sequence : a list of nonnegative numbers that is the degree sequence of some simple graph

ExampleExample

A recursive condition

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The lists 1,0,1 and 2,2,1,1 are graphic

The list 2,0,0 is not graphic

ExampleExample21

The list 33333221 is graphic

33333221w2223221

3222221v111221

221111u10111

11110

The realization is not unique!

u

v

u

v

u

w

Graphic sequencesGraphic sequences22

Graphic Theorem:For n > 1, the nonnegative integer list d of size n is graphic if and only if d’ is graphic, where d’ is the list of size n-1 obtained from d by deleting its largest and subtracting 1 from its next largest elements. (The only 1-element graphic sequence is d1=0)

DigraphsDigraphs23

A directed graph or digraph G is a triple consisting of a vertex set V(G), and edge set E(G), and a function assigning each edge an ordered pair of vertices

Tail: the first vertex of the ordered pairHead: the second vertex of the ordered pairEndpoints: tail and headAn edge: from tail to head tail head

DigraphsDigraphs

Loop: an edge whose endpoints are equalMultiple edges:

◦edges having the same ordered pair of endpoints.Simple graph:

◦each ordered pair is the head and tail of at most one edge◦One loop may be present at each vertex

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DigraphsDigraphs

In a simple graph◦An edge uv: tail u and head vFrom u to v

◦v is a successor of u◦u is a predecessor of v

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u v

ApplicationApplication

Finite state machine

Markov chain

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DD- UD+

DU+ UU-

DD+ UD-

DU- UU+

G B

.2

.3

.7

.8

DigraphsDigraphs

Path◦A simple digraph whose vertices can be linearly ordered so

that there is an edge with tail u and head v if and only if v immediately follows u in the vertex ordering

Cycle◦Defined similarly using an ordering of the vertices on a

circuit.

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ExampleExample

Functional digraph of f◦The simple digraph with vertex set A and edge set

{(x,f(x):xA)}◦For each x, the single edge with tail x points to the image of

x under f.Permutation

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7

1

2

4

3 5

6

001010010100100001111111

011110110101101011

DigraphsDigraphs

Underlying graph 相關圖 of a digraph D◦The graph G obtained by treating the edges of D as

unordered pairs◦The vertex set and edge set remain the same◦The endpoints of an edge are the same in G as in D◦But the edge become an unordered pair in G.

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D G

Example Example 30

ab

cdx

y z

ab

cdx

w

y z

0100

1021

0201

0110

z

y

x

w

zyxw

10000

11110

01101

00011

z

y

x

w

edcba

A(G) M(G)

0000

1010

0101

0100

z

y

x

w

zyxw

A(D)

10000

11110

01101

00011

z

y

x

w

edcba

M(D)

DigraphsDigraphs

Weakly connected◦Underlying graph is connected

Strongly connected (strong)◦For each ordered pair u,v of vertices, there is a path from u

to v.Strong components

◦Maximal strong subgraphs

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Example Example 32

x y

a b c d e

a b c d e

Not strongly connected

5 strong components

1 strong component

3 strong components

Degrees and digraphsDegrees and digraphs33

Out-degree : d+(v) v is tail. (out-neighborhood N+(v) )

In-degree : d-(v) v is head. (in-neighborhood N-(v) )

Minimum in-degree: -(G)Maximum in-degree:Δ-(G)Minimum out-degree: +(G)Maximum out-degree: Δ+(G)

PropositionProposition

In a digraph G,

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)()()()()(

vdGevdGVvGVv

Eulerian DigraphsEulerian Digraphs

Eulerian trail◦A trail containing all edges

Eulerian circuit◦A closed trail containing all edges

Eulerian◦A digraph is Eulerian if it has an Eulerian circuit

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LemmaLemma

If G is a digraph with +(G)1, then G contains a cycle. The same conclusion holds when -(G)1.

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uMaximal path Pv u

Theorem Theorem

A digraph is Eulerian if and only if d+(v)=d-(v) for each vertex v and the underlying graph has at most one nontrivial component.

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ApplicationApplication

De Bruijn cycles◦2n binary strings of length n◦Is there a cyclic arrangement of 2n binary digits such

that the 2n strings of n consecutive digits are all distinct?For example:

◦n=4◦0000111101100101 works

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00000001001101111111111011011011

0

1

00

0

11

1011

0

0

01

1 01101100100100100101101001001000

Example Example 39

1o

ooo

o

o

1

1 1 11

1

o

o

001

000

011

010

100

1

110

111101

D4

TheoremTheorem

The digraph Dn is Eulerian, and the edge labels on the edges in any Eulerian circuit of Dn from a cyclic arrangement in which the 2n consecutive segments of length n are distinct.

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ExampleExample 41

00000001001101111111111011011011

0

1

00

0

11

1011

0

0

01

1 01101100100100100101101001001000

1o

ooo

o

o

1

1 1 11

1

o

o

001

000

011

010

100

1

110

1111010

12

3 4

56

7 8

9

10

11 12

1314

15

01234567

89101112131415

Degrees and digraphsDegrees and digraphs42

An orientation of graph G: a digraph D obtained from G by choosing an

orientation (xy or yx) for each edge xyE(G).

An orientation graph is an orientation of a simple graph

tournament 比賽 : complete graph and each edge with orientation.

ExampleExample

Consider an n-team league where each team plays every other exactly once.◦For each pair u,v

Include the edge uv if u wins Include the edge vu if v wins

At the end◦There is an orientation of Kn

◦The score of a team is its outdegree

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Exercise 1.3.8Exercise 1.3.8

Which of the following are graphic sequences? Provide a construction or a proof of impossibility

for each◦(5,5,4,3,2,2,2,1)◦(5,5,4,4,2,2,1,1)◦(5,5,5,3,2,2,1,1)◦(5,5,5,4,2,1,1,1)

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Exercise 1.4.19 or 1.4.20Exercise 1.4.19 or 1.4.20

A digraph is Eulerian if and only if d+(v)=d-(v) for each vertex v and the underlying graph has at most one nontrivial component.

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