ON EDGE-CRITICAL GRAPHS AND THE NOTION
OF VERTEX INDEPENDENCE IN GRAPHS
by
VINCENT EDWARD FA TICA
B.S., Syracuse University, 1972
M.S., Syracuse University, 1977
M.Ph., Syracuse University, 1985
ABSTRACT OF DISSERTATION
Respectfully submitted to the Graduate School of Syracuse University in
partial fulfillment of the requirements for the degree
of Doctor of Philosophy in Mathematics
August 1990
/
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ABSTRACT
Chapter one provides an introduction to and an overview of the work. The overview is condensed in this abstract. For the convenience of the reader, the basic notions and terms of graph theory are presented in chapter two.
Chapter three gives an account of most of the major known results on graphs which are edge-crticial with respect to independence number. Chapter four presents three characterizations of these edgecritical graphs.
The following is shown in chapter five. For the task of decomposing a graph into disjoint subgraphs so that each inherits a maximum independent set of vertices from such a set in the given graph, so that the sum of their independence numbers equals that of the given graph, and so that none of them may be further decomposed, the edge-critical graphs are necessary and sufficient to serve as the decomposing subgraphs.
In chapter six it is observed that two of the Platonic graphs are not decomposable in the sense of chapter five. Decompositions of the other three are given, that of the dodecahedron being particularly striking. Chapter seven presents general constructions yielding connected edge-critical graphs.
In chapter eight the following is shown. For the task of covering a graph with subgraphs so that the sum of the independence numbers of the subgraphs equals that of the given graph, the edgecritical graphs are necessary and sufficient to serve as the covering subgraphs. Several conjectures regarding generalized covering theorems and edge-critical graphs are made.
Chapter nine gives numerous technical results aimed at proving one of the conjectures of chapter eight, and allowing for an elementary and graph theoretic proof of an extension, due to Gerards, of a celebrated theorem of Konig. Chapter ten presents this proof, and gives an account of the progress made in the attempt to prove the conjecture mentioned above. This conjecture, if verified, would further extend Gerards' result.
Chapter eleven gives a useful corollary to a theorem of Graver and Yackel on the independence number of a tree, and points out its relation to the preceding work.
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ON EDGE-CRITICAL GRAPHS AND THE NOTION
OF VERTEX INDEPENDENCE IN GRAPHS
by
VINCENT EDWARD FA Tl CA
B.S., Syracuse University, 1972
M.S., Syracuse University, 1977
M.Ph., Syracuse University, 1985
DISSERTATION
Respectfully submitted to the Graduate School of Syracuse University in
partial fulfillment of the requirements for the degree
of Doctor of Philosophy in Mathematics
August 1990
Approved --------------
Date
( \
Copyright 1990
VINCENT EDWARD FA TICA
( ii
To my parents
John Joseph Fatica
and
Jean Moore Fatica
and to my grandmother
Carolyn Bertha Moore
in memory of her
iii
Contents
Chapter page
1. Introduction and Overview ....................................................................... 1
2. Graph Theory ......................................................................................... 5
3. The edge-critical graphs ........................................................................ 10
4. Characterizations of connected edge-critical graphs ................................ 16
5. Decompositions of Graphs ...................................................................... 20
6. The Platonic graphs .............................................................................. 24
7. Some methods for constructing edge-critical graphs ................................ 30
8. Coverings of Graphs ............................................................................. 42
9. Various Results .................................................................................... 49
10. Gerards' Extension of Konig's Theorem .................................................. 75
11. A note on a theorem of Graver and Yackel ............................................ 82
iv
1
CHAPTER 1 - Introduction and Overview
The subject of this thesis is graph theory. For the benefit of the reader
who is unfamiliar with the basic notions and terms of the subject, these are
provided in the next chapter. For more in-depth presentations of the rudiments
of graph theory, the reader is referred to books by Berge ([B]), Bondy and
Murty ([BM]), Harary ([H2D, and Graver and Watkins ((GW]). The treatment
given in the last of these is particularly encyclopaedic and places great
emphasis on its mathematical rigor and its efficient use of notation and
symbolism. For the purposes of this introduction and overview, it is presumed
that the reader is familia.r with graph theory.
As mentioned above, chapter 2 presents the basic notions and terms of
graph theory. Chapter 3 introduces the notion of edge-critical graph. Edge
critical graphs, per se, were studied for a brief time, by an apparently small
number of researchers, among them such notables as Lovasz and Gallai. The
collection of literature on edge-critical graphs is quite compact, and the
theorems presented in chapter 3 are indeed nearly all of the major results in
the area. Some of these are used to advantage later in this work. The others
are inlcuded in chapter 3 for their historical significance.
In chapter 4, three characterizations of edge-critical graphs are given.
None are to be found in the literature. Later in the work, two of these are put
to some use.
Chapter 5 asks and answers the following vaguely stated question: "Can a
graph be taken apart, or decomposed, into smaller, perhaps more manageable
(
2
pieces, in such a way that the phenomenon of vertex independence in it may
somehow be understood in terms of the same phenomena in these smaller
pieces?" This question is made precise, and it is shown that the edge-critical
graphs are, in a very natural way, both necessary and sufficient to serve as
the "pieces".
In chapter 6, we will look at a. few familiar graphs in light of the
developments of chapter 5. We will consider the graphs formed by the
vertices and edges of the Platonic solids: the tetrahedron, the octahedron, the
cube, the dodecahedron, and the icosahedron, and take them apart (as much as
we can) in the manner of chapter 5. The view of the dodecahedron so obtained
is particularly striking.
In chapter 7 are presented constructions of edge-critical graphs. They
yield arbitrarily large graphs, with arbitrarily large deficiency (a parameter
according to whose value edge-critical graphs are classified), and which are
highly connected.
In chapter 8, we turn our attention to coverings of graphs, a notion
closely related to that of decompositions of graph. We ask: "Can we 'cover' a
graph by subgraphs in such a way as to gain information about vertex
independence in it from considering vertex independence in these subgraphs?"
The question is precisely posed, and it is shown that in a certain very natural
sense, the edge-critical graphs are necessary and sufficient to serve as the
covering subgraphs.
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3
While still in chapter 8, we introduce the reader to a celebrated theorem
of Konig and observe that it can be easily formulated in terms of edge-critical
graphs. Proposition 8.4 gives a "sweeping" extension of Konig's theorem which,
because its hypotheses are so strong, is, unfortunately, of little or no practical
value. However, several stronger (and quite reasonable) conjectures are made.
Chapter 9 has a two-fold purpose. We embark on an attempt at proving a
conjecture (9.2), which, though it is merely the first unresolved instance of a
more general conjecture from chapter 8, would, if it were determined to be true,
provide an extension of Konig's theorem which goes beyond those known to
date. These efforts unfortunately do not come to fruition but they do proceed
a long way in a direction which it is hoped the reader agrees is worthy of
continued investigation.
At the same time as we are trying to prove the above-mentioned
conjecture, however, we are accumulating results sufficient to allow us to give
an elementary proof of an extension of Konig's theorem due to A.M.S. Gerards.
Gerards' theorem is weaker than our conjecture 9.2 (indeed, it would be an
immediate consequence of 9.2), and his proof uses advanced notions and
techniques, most notably those of linear programming and matroid theory. The
elementary proof presented here will make the proof of this extension of
Konig's theorem accessible to a greater audience.
Chapter 11 may be considered an epilogue. It deals with a notion
communicated to this author privately by Jack Graver. It is precisely this
notion which proved to be the seed from which all of these researches grew.
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4
The material presented in chapter 11 is not without connection to the
earlier work (indeed we'll see a rather particular instance of the kind of
decomposition mentioned in chapter 5). Also, the main result of chapter lJ,
while merely a corollary to a theorem of Graver and Yackel, is interesting in
itself. Finally, it is hoped that the inclusion of this material will serve as a
small gesture of this author's appreciation for the inspiration, assistance, and
encouragement that Jack Graver has provided him.
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5
CHAPTER 2 - Graph Theory
We shall use the simple term "graph" to connote what should be more
properly termed "finite, simple, undirected graph without loops". For our
purposes, a ~ G consists in a finite set V(G) (V, for short), the elements of
which are called the vertices of G, paired with another set E(G) (E for short),
whose elements, called the edges of G, are doubleton subsets of V. The sets V
and E, aside from the stipulations that V be finite, and that each element of E
be a doubleton subset of V, are arbitrary. We shall often write G = (V, E) to
denote the graph G whose vertex set is the set V, and whose edge set is the
set E.
For a pair u E V, v E V of vertices of a graph G = (V, E), if e = {u, v} E
E, (i.e., e is an edge of G), we shall say that the vertices u and v are adjacent,
or that u is adjacent to v, and vice versa. We shall also say in this case, that
the vertex u (or v), is incident with, or is an endpoint of, the edge e; we shall
also say that u and v are neighbors. Edges sharing a common vertex are said
to be adjacent. The notion of graph itself and this language that we have
adopted, make it possible for us to put to very good use our geometric
intuition, for these objects can so easily be "pictured". As an example,
consider the graph K4 = (V, E) where V= {a, b, c, d} and E = {{a, b}, {a, c}, {a, d},
{b, c}, {b, d}, {c, d}}. (The name, K4, of this graph, a graph which will be highly
significant throughout this work, will be explained below.) K4 may be pictured
as i 11 Ill ustra ti on 1:
6
Illustration 1: Two views of the graph K4 •
d b
In the Illustration, the heavy dots labelled a, b, c, and d represent the
vertices of K4 , and a line drawn between two dots reflects the fact that the
vertices represented by these dots comprise an edge of K4 •
K4 is just one example of a complete ~ that is, a graph in which
every possible pair of vertices comprises an edge. Other complete graphs (named
K11 K2 , etc. according to their numbers of vertices) are depicted in Illustration 2.
Illustration 2:
l l<:i_ Ks
Notice that in the case of K5 , that the lines drawn to represent the edges
were allowed to cross (their crossing is in fact of necessity, though a proof of
that fact is not within the scope of this work; the interested reader should
consult (CW]). It will often be necessary, or desirable, to do this. It is
understood that at the point where these lines cross, there is no vertex (there
not being a heavy dot there), and it should be imagined that the "edges" do not
in fact meet.
7
The graphs Kn (n = 1, 2, .. .) are special in that their edge sets are
"complete". Another set of graphs is the collection of cvcles (or circuits), C3 ,
C4 , ••• (for so named obvious reasons), some of which are pictured in Illustration
3. The length of a cycle is the number of edges in it.
Illustration 3:
C3 . Cs
Before we can give an adequate definition of these graphs, we need to
consider a couple of new notions. By the valence (or degree) of a vertex of a
( graph is meant the number of edges incident with it; for a given vertex v of a
graph, its degree is denoted p(v). Furthermore, a graph is call connected, if for
every pair u and v of vertices of a graph G, there exists some sequence
u 0 , u1' .. ., Un of vertices of G for which u 0 = u, Un = v, and {uH ui+1} is an edge
of G for each i = O, 1, ... , n - 1. Such a sequence is called a path from u to v.
Geometrically speaking, a graph G is connected if it is possible to get from any
vertex to any other vertex by "travelling" along the edges of G.
All of the graphs mentioned and depicted so far are connected. The
cycles C3 , C 4 , • • • may now be easily defined as the collection of connected
graphs for which every vertex has degree 2. The graph G = ({a, b, c, d}, {{a,
b}, {c, d}}) depicted in Illustration 4 is an example of a graph which is not
connected. (
8
Illustration 4: A graph which is not connected
0.. ··---------· b
c d Later in this work, we shall come across more refined notions of
connectedness. Presently, we need to explore a few more of the fundamental
notions and relations of graph theory.
As mentioned earlier, those vertices adjacent to a given vertex, v, are
called the neighbors of v; accordingly, the set of neighbors of v is called the
neighbor set of v and is denoted N(v).
A graph H = (V(H), E(H)) is said to be a subgraph of the graph
G = (V(G), E(G)) if (i) V(H) C V(G), and (ii) E(H) E(G) n P2(V(H)), where
P2(V(H)) denotes the collection of doubleton subsets of V(H). Certain types of
subgraphs of a given graph will be of particular importance. If G = (V, E) is a
graph and v E V, then G - v will denote the subgraph of G resulting from
deleting from V the vertex v, and deleting from E those edges incident (in G)
with v. Similarly, if G = (V, E) and S C V, then G - S will denote the
subgraph of G resulting from deleting from V the set S of vertices, and
deleting from E those edges incident with an element of S. And too, if G = (V,
E) and F C E, then G - F, (G - e if F = {e}), will denote the subgraph of G
resulting from the deletion from E of those edges in F.
If a graph G has as a subgraph a cycle C then by a chord of C is meant
9
an edge of G not on C, but incident with two vertices of C.
In a graph G = (V, E), a set I C V is said to be independent if no two of
its members are adjacent. Such a set whose cardinality is a maximum among
the cardinalities of all independent sets of vertices of G is called a maximum
independent set (of vertices of G), and the cardinality of a maximum
independent set is called the independence number of G, and is denoted o:(G). In
keeping with this notation, and for the sake of conciseness, a maximum
independent set will hereinafter be called an a-set of G.
A few other numbers associated with a graph G CV, E) are of interest:
(i) v(G) = IVCG) I
(ii) e(G) IE(G) I, and
(iii) 6'(G) v(G) - 2o:.(G), (v - 2a. for short), which has by a few authors been
called the deficiency of G.
In [Gl], A. George has written:
"Finding a largest set of mutually non-adjacent ... points
[an a-set, to us] is one of the oldest problems of graph theory.
This problem arises in many different contexts (e.g. chessboard
problems, coding theory, network theory, matching problems, etc.)."
It is precisely this notion of independence to which the remainder of this
work is dedicated.
10
CHAPTER 3 - The edge-critical graphs
In our investigation of vertex independence in graphs, we are asking a few
basic questions, which will be formulated more precisely later:
(i) Is it possible to take apart a given graph G in such a way that we may
gain information about the phenomenon of vertex independence in G by
studying the same phenomenon in each of the pieces?
and, if this is the case
(ii) Can we restrict our attention to any particular, hopefully small and well
defined, class of graphs to serve as these "pieces"?
and, if this too is the case,
(iii) What can we say about this particular class of graphs?
Before attacking these questions, we need to be introduced to a particular
class of graphs, the edge-critical graphs.
Definition: A graph G
every edge e of G.
(V, E) is said to be edge-critical if cx(G - e) > cx(G) for
That is, G is edge-critical if the removal of any one of its edges causes
the independence number to increase.
The edge-critical graphs were first considered by Zykov ([Z]) in 1949, and
the list of those who have studied them since then is short, though it includes
some of the most prominent names in graph theory. And the body of literature
on edge-critical graphs is consequently small and compact.
11
We will present below many (in fact, almost all) of the basic results on
edge-critical graphs. A few, mainly more technical, results will be saved for
later chapters. Fairly thorough accounts of the major results on edge-critical
graphs may be found in [G1J, [H2J, and [L2].
3.1 (see [B], p. 287) Let G = (V, E) be a connected edge-critical graph,
and let U V have the property that G - U is not connected (such a set U of
vertices is called a cut-set of G). Then the subgraph of G ~anned by the
vertices of U (i.e. G - (V - U)) is not a complete graph.
3.2 (due to Berge, see [BJ, p. 286) In an edge-critical graph, any pair of
adjacent edges lies on a chordless cycle of odd length.
3.3 (due to Hajnal, see [H1J) If G is an edge-critical graph without
isolated vertices (ie: without vertices of degree 0), then for every independent
set S of G, IN(S) I ~ IS J. [ N(S) = U N(V) ] vES
3.4 (ibid) If G is an edge-critical graph without isolated vertices, then
for every vertex v of G,
p(v) ~ v(G) - 2cx(G) + 1 = o(G) + 1.
3.5 (a corollary to 3.4, due to Erdos and Gallai, see [B], p. 293) An edge-
critical graph G contains at least 2cx(G) - v(G) isolated vertices.
As the reader may be coming to realize, the parameter o(G) = v(G) - 2cx(G)
is of great importance in classifying the edge-critical graphs. We shall adopt
the following notation:
r = the collection of all edge-critical graphs
r n = the collection of edge-critical graphs G for which
o(G) = v(G) - 2a.(G) = n.
A = the collection of connected edge-critical graphs
l:ln = A n r n· (n = 0, 1, 2, ... ).
12
The fact that a connected graph has no isolated vertices, together with 3.5
above provide the range of values of n in the definition of An.
For convenience, we will let ;:1_ 1 denote the collection of graphs whose
only member is the graph K10 the graph with one vertex and no edges. (This
graph is edge-critical and connected, both trivially, and indeed o(K 1) = -1).
The next three results are of particular importance in the remainder of
this work.
3.6 (due to Andrasfai, see [A], p.10) A0 is the collection whose only
member is K2, the graph with two vertices and one edge.
3.7 (ibid) A 1 = {C3 , C5 , C7, ... } i.e., the collection of cycles of odd length.
A graph is called an odd subdivision of K4 or, to us, a very-odd-K4 if it
can be obtained from the graph K4 by replacing some or all of the six edges of
K4 by mutually disjoint paths of odd length. Illustration 5 depicts some very
odd-K4's.
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Illustration 5:
3.8 (ibid) t.. 2 is the collection of graphs consisting solely of the very-odd-
K/s.
Though it is not the last in our list, investigation into the edge-critical
graphs seems to have come to an end with the following finite basis theorem.
3.9 (due to Lovasz, see [Ll], p. 726) For each of 6 = 1, 2, 3, ... there
exists a finite collection of graphs in A 8 with the property that all the graphs
in t..0 can be obtained from these by the replacement of some or all of their
edges by mutually disjoint paths of odd length.
Note: We've seen in 3.7 that the single graph K3 provides such a "finite basis"
for A1, and in 3.8 that the single graph K4 provides one for .6. 2 • Such
simplicity however does not remain the rule. It is known that for 6 > 3 the
number of graphs needed is greater than one. The constructions given later in
chapter 7 provide two such examples for each 6 > 3.
Lovasz finishes the proof of 3.9 by deducing that a graph in the "basis" 8
for a given o may have no more than 22 vertices. This bound is not believed
to be sharp. In the case 6 = 3, this bound is approximately 1.84 X 1019, In
(S3], Suranyi gives a better bound, namely 57.
14
We conclude this chapter with two theorems which will be valuable to us
later.
As 3.8 and 3.9 suggest, this process of odd-subdivision (and its "reversal")
preserve the property of edge-criticality. This fact is stated formally in the
following theorem, due to Andrasfai:
3.10 (see [A], p. 14)
a) If G = (V, E) E. f and v E. V with p(v) = 2, v having neighbors u and
w, then the graph G' obtained from G by the deletion of the vertex
v and the edges {u, v} and {v, w}, and the subsequent identification
of the vertices u and w, is also in f; moreover, o(G') = o(G).
b) If G = (V, E) E. r, v E. V, and p(v) > 2, and if v is split into two new ver-
tices v 1 and v 2 , while N(v) is partitioned into the non-empty sets N(v 1)
and N(v2 ), and if a new vertex u, and the edges {u, v 1} and {u, v 2} are
added, then the resulting graph, G', is also inf; moreover o(G') = o(G).
Before we can appreciate the following beautiful theorem of Gallai,
characterizing the edge-critical graphs which are connected, but not highly
connected, we must elaborate on the notion of connectedness itself.
Definition: A connected graph, G = (V, E), is said to be k-connected
(k = 1, 2, .. .) if for every S C V with ISi :;;; k - 1, G - S is also connected.
That is, G is k-connected if G has no cut-set containing fewer than k
vertices. By the connectivity of a graph, G, we mean the greatest integer k
for which G is k-connected.
15
Gallai's theorem, below, characterizes the edge-critical graphs of
connectivity 2 (i.e., the (2 but not 3)-connected ones). Note that 3.1 implies
that connected edge-critical graphs are 2-connected.
3.11 (due to Gallai, comminicated privately to Andrasfai, see [A], p. 13)
a) Let G1 and G2 be two graphs in A, having no vertices in common, and
each having more than one edge. Let e = {uJJ vJ be an edge of G1
and let x be a vertex of G2 • Let Gi be the graph G1 - e and let G2
be obtained from G2 by splitting x into two vertices u 2 and v 2 while
partitioning N(x) into the non-empty N(u 2 ) and N(v 2). Let G be the
graph obtained by identifying u 1 and Vu of G~ with u 2 and v 2 of G2
(respectively). Then G E A.
b) Let G E A and suppose {u, v} is a cut set of G. Then {u, v}
is not an edge of G (cf., 3.1) and the graph G - {u, v} has exactly two
components. Let N~' and N2' be the sets of vertices of these
components, and for i = 1, 2, let Ni = Ni' U {u, v}. Let Gi be the
subgraph of G spanned by N~ . Then exactly one of the graphs Gi has
the property that the graph Gi obtained from it by adding the edge
{u, v} belongs to A; and the other one (and only the other one) has
the property that the graph, Gi, obtained from it by identifying u and
v belongs to A.
c) In both a) and b)
v(G) = v(G 1) + v(G 2 ) - 1 and
cx.(G) = cx.(G 1) + cx.(G 2 ), and consequently
o(G) = 0CG1) + 0CG2) - 1.
16
CHAPTER 4 - Characterizations of connected edge critical graphs.
No characterizations of edge-critical graphs appear in the literature.
Three are given below. The two characterizations contained in 4.1 characterize
edge-critical graphs in terms of the local interaction between a-sets and some
particular sets of vertices. The third characterization, given in 4.2 [as will be
seen later] characterizes edge-critical graphs by a property which will lead
quite naturally to one of the next chapter's theorems on decompositions of
graphs in general.
4.1 - For a connected graph G, the following are equivalent.
(a) G E A;
··-·- tlr)- for every vertex v oT G, N(vJ is minimal wTffi r-espect To set indusion
among the subsets S of V(G) with the property that S n I ~ 0 for
every O'.-set I of G - v;
(c) for every vertex v of G, and for each vertex x E N(v), there exists
an o.-set I of G with the property that I n N(v) = {x}.
Proof: (a implies b) Suppose G E A and let v be a vertex of G. If N(v) = 0,
then, being connected, G is the graph K1' whose only vertex is v; in this case
(b) follows trivially. Suppose N(v) ~ 0 and let u E N(v) and let e = {u, v}.
Let I be an O'.-set of G - e. II I > a(G) since G is edge-critical; in fact:
(i) contains both u and v, for if not, I would be independent
in G, which is absurd. I - {v}, being independent in G, is
consequently an a.-set of G and thus cx(G - v) = cx(G).
(ii) Every o:-set of G - v meets N(v), for if one, say J, did not, J U {v}
would be an independent set in G with
IJ u {v}I !JI + 1 = cx(G - v) + 1 = <x(G) + 1
which is absurd. And,
17
(iii) N(v) has the desired minimality. To see this suppose S is a proper
subset of N(v) and let w E N(v) - S. Let e denote the edge {v, w} of
G. Let I be an a.-set of G - e. II I = cx(G) + 1 (as in (i)). Moreover,
I ll S = 0 since v E I (as in (i)). Thus, I - {v}, which has cardinality
a(G) = a(G - v) is an a-set of G - v which fails to meet S.
(b implies c) Suppose condition (b) holds and let v be a vertex of G. If
N(v) = 0, then (c) is vacuously true. So suppose w E N(v). Let e denote the
edge {v, w} of G and let I be an ex-set of G - e. I n N(v) = {w}, for I is
independent and contains v. This is as desired.
Cc implies a) Suppose condition (c) holds. Let e = {v, w} be any edge of
G. Let I be an a.-set of G with I n N(v) = {w}. I U {v} is independent in G - e
and thus
a(G - e) :;:;:; II U {v} I II I + 1 = a(G) + 1
implying G E t:... 0
The following definition will facilitate the statement and proof of our
third characterization of connected edge-critical graphs.
(
(
18
Definition: A subgraph H of a graph G is called hereditary if for every o:.-set
I of G, I n V(H) is an o:.-set of H.
4.2 A connected graph is edge-critical if and only if it has no proper
hereditary subgraphs.
Proof Suppose a connected graph G is edge-critical and suppose the proper
subgraph H of G is hereditary.
Case 1: If V(H) = V(G), then o:.(H) = o:.(G) for if I is an a-set of G,
I n V{H) = I. Now, H being a proper subgraph of G, there must be an edge e of
G which is not an edge of H. Let I be an a-set of G - e. Since G is edge
critical, III = a(G) + 1 which is the same as a(H) + 1. But I is independent in
H. This is absurd.
Case ~: If V(H) ~ V(G), let K be the subgraph of G spanned by the vertices in
V(G) - V(H). Since H is hereditary, for every O'.-set I of G, II n V(H) I = ryJH)
and so for every a-set, I, of G, II n V(K)I = a(G) - a(H).
Let e be an edge connecting a vertex v of H to a vertex w of K and let
be an a-set of G - e. Since G is edge-critical, II I = cx(G) + 1. Certainly
cannot contain more than a(H) vertices of H. On the other hand, if II n V(H) I
::;; cx(H), then I - { v} is an ex-set of G with
!CI - {v}) n V(H)I = II n V(H)I - 1 ::: cx(H) - 1
contrary to the supposition that H is hereditary.
19
On the other hand, suppose G is connected and not edge-critical. Let e be
an edge of G with the property that a(G - e) = o(G). Then, G - e is itself a
proper hereditary subgraph of G. 0
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20
CHAPTER 5 - Decompositions of Graphs
We now turn our attention to the questions posed at the beginning of
chapter 3. We shall formulate the first two of them more precisely, and
answer them.
Definition: Let G = (V, E) be a graph. The subset Va.CG) of V is defined to be
the union of all the a_-sets of G. Va.CG) is called the a_-support of G.
Definition: By an o:.-decomposition of a graph G, we mean a collection
{Hu Hz, ... , Hk} of subgraphs of G with the following properties:
(a) each H, is a hereditary subgraph of G,
k (c) U VCH;) = Va.CG).
i=l
Note: Of great importance is that the properties (a), (b), (c) above imply that if
k {Hu Hz, ... , Hk} is an cx.-decomposition of G, then a(G) = 2:cx.(Hi).
i=l
Definition: By the trivial a-decomposition of G, we mean the collection {G}.
Definition: An a-decomposition of a graph G will be called a preferred
a-decomposition if none of. the Hi have non-trivial a-decompositions.
Note: If {H1, Hz, ... , Hk} is a preferred ex-decomposition, then each H; is
connected.
(
21
5.1 If G is connected and edge-critical, then G has no o.-decomposition other
than the trivial one.
Proof: 4.2 characterizes connected edge-critical graphs as those connected
graphs having no proper hereditary subgraphs. From this alone, the desired
result follows. D
5.1 tells us that if we desire a collection of graphs, the members of which
suffice to serve as the "building blocks" for the kind of decomposition
mentioned above (i.e., to serve as the decomposing subgraphs), then we are
obliged to include in that collection the connected edge-critical graphs. That
is, that they are necessary. We shall soon see that they are sufficient as well.
5.2 If G is not edge-critical, then G has a non-trivial ex-decomposition. (Note
that only when G is connected and V O'JG) = V(G) is {G} an ex-decomposition.)
Proof: First note that if the subgraph G0 of G spanned by the vertices in
VaCG) is edge-critical, then the components of G0 constitute a (preferred) ex
decomposi ti on.
If G0 has no edges, then G0 is a collection of isolated vertices and so is
edge-critical, a case which we have already considered.
So let e0 be an edge of G0 with the property that ex(G0 - e0 ) = ex(G0 ), and
let G1 = G0 - e0 • Continue in the following way: having constructed Gi, with
V(Gi) = V(G 0 ) and ex(Gi) = ex(G 0 ), if Gi is not edge-critical, let ei be an edge of
Gi with the property that o:.(G, - ei) = ex(Gi) and let GH 1 = Gi - e 1• This cannot
continue ad infinitum. Eventually, say when i = N, GN will be edge-critical,
with V(GN) = V(G 0 ) and o:(GN) = ex(G). At this point, the components of GN
(which are each edge-critical) constitute a (preferred) ex-decomposition of G. D
5.3 (Lemma) If H is connected and edge-critical, then there exists no
collection H1, ••• , Hk of disjoint, connected subgraphs of H, the union of whose
vertex sets span H, and which has the property that o:.(H) = 2:)x(Hi).
Proof: Let H be connected and edge-critical and suppose that such a collection
{H1' ... , Hk} does exist. Let e be an edge of H joining vertices in different H1•
Since H is edge-critical, ex(H - e) = ex(H) + 1. Let I be an o:.-set of H - e. Then
I is independent in the subgraph of H whose components are the Hi. But
III = ex(H) + 1 = 2:exCH) + 1, which is absurd. D
( \
The following is almost immediate:
5.4 If a graph G has an ex-decomposition whose elements are connected edge-
critical subgraphs, this ex-decomposition is a preferred ex-decomposition.
Proof: If {H1, ••• , H1G} were an a-decomposition of G with each H1 connected and
edge-critical, while not being a preferred ex-decomposition, this would imply the
existence, for at least one of the Hit of a collection of subgraphs, the existence
of which is precluded by 5.3. 0
5.5 Every graph G has a preferred ex-decomposition whose elements are edge-
critical subgraphs of G.
23
Proof: By 5.1 and 5.4, if G is edge-critical, the components of G, which are
also edge-critical, comprise a preferred a-decomposition of G.
If G is not edge-critical, then the construction used in the proof of 5.2
yields an a-decomposition whose elements are connected edge-critical subgraphs.
By 5.4, this is a preferred a-decomposition. 0
This shows that the connected edge-critical graphs are sufficient for the
task of decomposing graphs in the sense of what we've called a preferred
cx-decomposi ti on.
24
CHAPTER 6 - The Platonic graphs
The Platonic solids have been of interest for a very long time, and more
recently, their vertices and edges have become interesting examples of graphs.
We digress here to view these well known graphs in light of the foregoing
remarks. (Note that for each of these graphs, Va(G) = V(G).)
1. The tetrahedron, depicted in Illustration 6, is an edge-critical graph.
Indeed, it is the graph K4, the scle building block of the class .6. 2 of
connected edge-critical graphs.
Illustration 6: Tetrahedron (K4 )
2. lllustration 7 depicts the octahedron.
(
Illustration 7: Octahedron
I
I \ I
25
In the illustration, all the edges, both solid and broken, are edges of the
octahedron. The two subgraphs (triangles, three-cycles) show a preferred
a-decomposition of the octahedron into two edge-critical subgraphs, each with
6 = 1. The decomposition is unique up to automorphism.
3. Illustration 8 depicts the cube (solid and broken edges) together with a
preferred a-decomposition (solid edges only) into 4 edge-critical subgraphs,
each with o = 0.
26
Illustration 8: The cube
f': //1 ' I "
/ I /
I " /
"- /
~ ( V=- 8
I o\ =- 4 I d = 0
;b ~ / ' " / " I / " I
/ " k ~
( 4. The icosahedron, shown in Illustration 9, is, as will be shown, an edge-
critical graph.
(
Illustration 9: The icosahedron
v:: /,;),_
o<. :. 3
d =- b
27
Let us first see that for the icosahedron, ex = 3. Referring to the
labelling in Illustration 9, if an independent set of vertices
(i) contains 0 or 1 of the x's, it may contain at most 1 of the v's and at
most 1 of the y's;
(ii) contains 2 of the x's, then, of the v's and y's, it may contain none of
one and at most 1 of the other;
(iii) contains 3 of the x's, it may contain none of the v's and none of the y's.
In any case, the cardinality of an independent set of vertices in the icosahedron
is limited to 3, and since {x2 , x4, x6 } is, in fact, independent, ex = 3.
In showing that the icosahedron is edge-critical, we need only check the
(
28
removal of the edge {xl> x2}, for, by virtue of the symmetry of the icosahedron,
this edge may represent any of its edges. Upon removing {x1, x2 ), the following
set of four vertices is independent: {x 11 x2 , v 2 , y 2}.
Thus, the icosahedron is edge-critical. Moreover, o v - 2<X 12 - 2(3) =
6.
5. The most striking of these examples is the dodecahedron, depicted in
Illustrations 10 and 11. Illustration 10 gives a typical depiction of the
dodecahedron while Illustration 11 shows a preferred o:.-decomposition of it
into two edge-critical subgraphs, each with o = 2. (Recall that ~2 is the
collection of odd-subdivisions of K4 , or as we have called .them, the very
odd-K4's.) The labellings of the vertices in the two illustrations are
consistent.
Illustration 10: The dodecahedron
V = .2..0 <:>(. :::. 8'
d = t
29
Ill ustra ti on 11: Pref erred o:-decomposi ti on of the dodecahedron
- - - - - -.- -- -_., ........
/
/ ' D - - - - ' - - '+ '
/
~ ' I '\
\ F ~ b
B ;;...
G- ... 7 ... \
\ 10 \
' l --' - - - /'
/
' '>/'
' _,. -..----- - - - :- - -
With this view of the dodecahedron, it is quite easy to demonstrate that
its independence number is 8. Each of the decomposing very-odd-K/s has o: = 4
(having v = 10 and o = 2). Thus for the dodecahedron, o: ~ 8. Since
{A, C, F, I, 1, 4, 7, 10} is indeed of cardinality 8 and independent in the
dodecahedron, its independence number is equal to 8.
(
(
30
CHAPTER 7 - Some methods for constructing edge-critical graphs.
Before continuing along more abstract lines, we will give a few
constructions which yield connected edge-critical graphs.
7.1 V(G) be minimal with respect to
set inclusion among the subsets of V(G) which have non-empty intersection with
every a-set of G. Let G' be the graph whose vertex set is
V(G') V(G) U {v},
where v is a "new" vertex, and whose edge set is
(i.e., add the new vertex v to G, and connect it with edges to each vertex in
U). Then,
G' E ll. and o(G') o(G) + 1.
Proof: cx(G') = cx(G) because U has non-empty intersection with every a-set
of G. Since G' is connected, we need only to show that the removal of any
edge from G' causes its independence number to increase.
If e is one of the edges of G, then o:.(G' - e) > o:.(G'), since G itself is
edge-critical. Now let e be one of the new edges, say, without loss of
generality, {v, v 1}. To conclude that a(G' - e) > a(G'), it will suffice to show
that G has an ex-set I with I n U = {v1}, for then I U {v} will be independent in
(
31
G' - e and of cardinality o:.(G) + 1 > o:.(G').
(The argument that follows is similar to the one presented in the proof of 4.1.)
Some a.-set of G contains v 11 for if not, the set U - {vJ would contradict
the minimality of U. And too, if every o:.-set of G which contains v 1 also
contains some other element of U, the set U - {v 1} would again contradict the
minimality of U. Thus G has an o:.-set meeting U in precisely {v 1}. This is as
desired, and the proof is complete. D
Sets such as the set U in the preceding theorem do not in general make
themselves obvious.
corollary to 7 .1.
One type, however, does as is seen in the following
7 .2 (Corollary to 7 .1) If G is edge-critical and connected and if v E V(G) then
the graph G' obt ined from G by the addition of a new vertex u, and new edges
from u to each of the members of {v} U N(v) is also connected and edge
critical.
Proof: The characterization of connected edge-critical graphs given in 4.l(b)
guarantees that the set {v} U N(v) has the necessary properties to serve as the
set U in the construction of 7 .1. D
Illustration 12 depicts how the construction of 7 .2 works in the particular
case of the complete graphs K1, K2 , K3 , •.• which are respectively elements of
A-1' A 0 , A 1, ••.•
Illustration 12: The construction of 7 .2.
0
1<.:i.
+
+
+
> •
Another example is given in Illustration 13.
Illustration 13:
+
7 .3 Let G have vertices v 11 v 2 , ... , v Zk+H y 1, ... , y k+H x, and let
32
0 • k~
i> }(3
l<.'f
(i) the vertices v 1, v 2 , ... , v 21,+1 be the vertices of a chordless cycle in
G of length 2k + 1,
(ii) Yi have an edge to v 2 i-l and v 2 i (i 1, ... , k),
(iii) Yk+l have an edge to v 2k+H
(iv) x have an edge to each Yi·
Then G c:: .C.k+i-
For k 3, the graph G is depicted in Illustration 14a.
Illustration 14:
v, Proof: v(G) = (2k + l) + (k + 1) + 1
Cb)
·v7 (x.)
3k + 3. o:.(G) = k + 1, for
33
(a) an ind.:::pendent set of G containing none of the Yi must contain x, and may
contain at most k of the vi, thus limiting its cardinality to k + 1,
(b) an independent set of G containing, say, j > 0 of the Yi may not contain x,
and may contain at most (k + 1) - j of the vi> again limiting its cardinality
to k + 1, and
(c) there is an independent set in G of cardinality k + 1, namely
6(G) = v(G) - 2o:.(G) = 3k + 3 - 2(k + 1) = k + 1.
To see that G is edge-critical, observe that
(a) if any edge {vi. v J} is removed, then an independent set of cardinality
k + 2 exists, containing k + 1 of the vi and containing x;
(b) if any edge {vi, y) is removed, then there exists an independent set of
cardinality k + 2 containing all the y /s and vi; and
(
(c) if any edge {x, y ;} is removed, there exists an independent set in G of
cardinality k + 2 containing k of the Vi (but no neighbor of y J), together
with YJ and x. D
34
The graphs resulting from this construction are (2 but not 3)-connected;
indeed, the vertex Yk+i has degree 2. Andrasfai's construction (3.10) could then
be used to delete Yk+i and identify its two neighbors. The graphs which result
(the case k = 3 is depicted in Illustration 14 (b)), while they are not more
highly connected, are, since they have no vertices of degree 2, necessarily
members of Gallai's "finite bases" for the various Ak.
7.4 Let G have vertices vl> v 2 , ••• , v 2k+H xi> x2 , ... , x4k+Z• (k ::: 1). Let the vi
be the vertices of a complete graph K2 k+l and let the x1 be the vertices of a
chordless cycle of length 4k + 2. Finally, let each v 1 have as neighbors (other
than the other v's), the vertices x2 i and XzHZk+l (the subscripted indices being
considered modulo 4k + 2). Then G E A2k+P
Illustration 15 depicts the graphs G for k 1 and k 2.
35
Illustration 15:
-x.,
'X-7
k::- I d =3 ) f<=;L cf =-S
\
Proof of 7.4: The set {x21 : i = 1, 2, ... , 2k + 1} is independent in G. Thus
o:(G) :?: 2k + 1. An independent set in G may contain no more than one of the
v's, and if such a set contains one of the v's, then it may contain only 2k of
the x's (since the inclusion of one of the v's precludes the inclusion of an
antipodal pair of x's). Thus cx(G) = 2k + 1. Since v(G) = (2k + 1) + (4k + 2) =
6k + 3, o(G) = (6k + 3) - 2(2k + 1) = 2k + 1.
G is certainly connected. To see that G is edge-critical, observe that
(a) if an edge between two of the x's is removed, say, without loss of
generality, {xll x4k+2}, then there exists an independent set of cardinality
2k + 2 containing 2k + 1 of the x's and one of the v's. As we have
supposed without loss of generality that {xll x 4k+2} was removed, this
independent set could be taken to be
(
36
for 2k + 1 of the x's appear, but not x2 and x2k+3 , the x's which neighbor v 1•
(b) If one of the edges {v1, v) is removed, we begin constructing an
independent set I of cardinality 2k + 2 by including in it v 1 and v J• The
inclusion of vi in I precludes the inclusion of an antipodal pair of the x's,
leaving as candidates for inclusion in I two strings of x's of even length
(length 2k to be precise). In any string of vertices of even length 2k, an
independent set of k of them may be chosen so as not to include any
given one of them. If we now include in I an independent set of k
vertices in each of these strings which fails to include the neighbors of
vJ, we have arrived at the desired independent set in G - {v;, v.1} of
cardinality 2k + 2.
(c) If one of the edges {xH v J} is removed from G we may find the desired
independent set of cardinality 2k + 2 by simply choosing vJ along with the
2k + 1 vertices xH XHz• xH4 , ••• , xH'l,k (indices modulo 4k + 2), noting that
the other neighbor of v J among the x's (which is XHzk+r) is not chosen. D
One may consider the icosahedron (see chapter 4) to be constructed in the
following way: Let G have vertices v1' v 2 , v 3 , xll x 2 , ... , x6 , YH y 2 , y 3 • Let the
v's (respectively, the x's, the y's) be the vertices of a triangle (respectively, a
chordless 6-cycle, a triangle). For i = 1, 2, 3, let Yi also have as neighbors
Xzi-1' Xw and XzHl• and for i = 1, 2, 3, let Yi also have as neighbors Xzi• Xzi+H
and x 2H 2 (the subscripted indices on the x's being considered modulo 6). This
is the icosahedron (pictured below).
37
Illustration 16:
V =I~
d ::. b
If we use the same method to attach two 4-cycles to an 8-cycle, we again
arrive at an edge-critical graph (see 7.5 below). This graph was also considered
by Watkins (see (W], p. 245) but for quite different reasons. The construction
in general, however, does not yield edge-critical graphs.
(respectively, the x's, the y's) be the vertices of a chordless 4-cycle
(respectively, 8-cycle, 4-cycle). Let v, also have as neighbors x 2i-1' Xw and
x2 H 11 and let y 1 also have as neighbors xw x2 Hi• and x2 H 2 (the indices on the
x's beng considered modulo 8). Then G E A8 •
(
(
38
Illustration 17:
Proof: We will first show that ex = 4. There are three cases to consider,
according to whether a given maximum independent set I of vertices in G
contains 0, 1, or 2 of the y's.
Case 1: If I contains none of the y's, then either
(a) I contains none of the v's, in which case I contains at most 4 of the
x's,
(b) I contains 1 of the v's, in which case I contains at most 3 of the x's,
or
(c) I contains 2 of the v's, in which case I contains at most 2 of the x's.
Case 2: If I contains 1 of the y's, then either
(a) I contains 3 of the x's and none of v's,
(b) I contains 2 of the x's and at most 1 of the v's, or
(c) I contains 1 of the x's and 2 of the v's.
(
(
Case 3: If I contains 2 of the y's, then either
(a) I contains 2 of the x's and none of the v's,
(b) I contains 1 of the x's and 1 of the v's, or
(c) I contains none of the x's and 2 of the v's.
39
In any case, the cardinality of I is limited to 4 and since {x 11 x3 , x5 , x7 } is
indeed independent, a. = 4.
In showing that G is edge-critical, we need to check the removal of only a
few of the edges, for the remainder of the cases follow from the high degree of
symmetry of G. The representative edges are:
(1) {y11 y 2}, representing a typical edge on either of the 4-cycles;
(2) {xl> x 2}, representing a typical edge on the 8-cycle;
(3)
(4) {yl> x 3}, representing typical edges from one of the 4-cycles to the 8-cycle.
The independent sets of cardinality 5 which arise upon the removal of
these representative edges are listed below, respectively:
(1) {yl> Y2, V11 V3, X3},
(2) {Xl> X2 1 X4, V3, y3},
(3) {y 1, x 2 , x6 , v 2 , v 4} and
(4) {y1' X3, X11 X5, y3}.
Lastly, o(G) v(G) - 20'..(G) 16 - 2(4) 8. D
The following very similar construction can be generalized, and yields
4-connected edge-critical graphs in A 4 , A 6 , A 8 , ....
40
7 .6 Let G have vertices x 1, x2 , .. ., x2k+i• y 1, y 2 , .. ., Yn+I> (k 2: 1). Let the x's
and the y's be each the vertices of a chordless odd cycle of length 2k + 1, and
let X; also have as neighbors y,_ 1, Yi, and Yi+i (subscripts modulo 2k + 1). Then
G E A2k+2·
[In 7 .6, letting k = 1 result$ in the complete graph K6 , which is in A 4 •
The results for k = 2 and k = 3 are depicted below.]
~I
k=::i. ) d = 6 j <>
1~ = 3 )
Proof of 7.~: Clearly, v(G) = 4k + 2. Now consider the edges {x1, y 1}, {x2, y 2},
.. ., {x2k+H Yzk+ 1}. For the same reason that a collection of k + 1 vertices on a
2k + 1 cycle must contain an adjacent pair, a collection of k + 1 vertices of G
must contain a representative of each of some "consecutive" pair of these
edges, say {xH y 1} and {xi+l> yi+ 1}. But the vertices comprising these two edges
are pairwise adjacent, implying that a collection of k + 1 vertices of G cannot
be independent. Thus a.(G) :::;; k. And since {x2, x 4, ... , x2 k} is independent in G,
41
and has cardinality k, a(G) k.
Therefore,
o(G) = v(G) - 2o:.(G) (4k + 2) - 2k 2k + 2.
The high degree of symmetry of G allows us to check only a few
representative edges of G in determining that G is edge-critical. They are:
(1) {x 1, x2}, representing an edge on either of the 2k + 1-cycles;
(2) {xu y 1} and
(3) {x1, y 2 }, representing an edge between the two 2k + 1-cycles.
The removal from G of each of these results in an independent set of k +
1 vertices. These independent sets are given, respectively, below.
(1) {x1, Xz, X4, X5, ... , X21J,
(2) {xll YH X3, X5, ... , Xn:-1},
(3) {x1' y 2• X4, X5, X3, .. ., Xzd.
So G is edge-critical, and, as desired, in .l2k+2· D
42
CHAPTER 8 - Coverings of Graphs.
The notion of a-decomposition of an arbitrary graph G, presented in
chapter 5, while abstractly appealing, has a certain shortcoming: it presupposes
some knowledge of the phenomenon of vertex independence in G, at the very
least, knowledge about the set V o.(G). The question of determining the set
V o.(G) will not be considered in this work, and it seems to be, in general, a
difficult one.
So we will consider a closely related notion, that of a covering of a given
graph G by subgraphs, in the hope that we may understand vertex independence
in G, with only a knowledge of some properties of G which are relatively
explicit (compared to some knowledge of V O'.(G)) in the definition of G itself.
Definition. By a covering of a graph G, we mean a collection {H 1, H2 , •.• , Hk} of
k connected subgraphs of G with the property that V(G) = U V(Hi).
i=l
k
For any covering of G, o:.(G) ~ L:a(Hi), for if I is an a-set of G, i=l
k k k ex.CG) = II I I .u o n vmi) I ~ L: II n VCH;) I ~ :Lex.CH;).
t=l i=l i=l
Definition. By an ex-covering of a graph G, we mean a covering of G with the k
additional properties that cx(G) = L:o:.CH;) and that the H; are vertex disjoint. i=l
1' That is, a covering by disjoint subgraphs for which L:o:.CH;) is a minimum. It
i=l
will be seen that with some knowledge of the kinds of subgraphs which G has,
we can put very definite limits on the subgraphs needed to serve in an
(
43
o:-covering.
The notions of O'.-decomposition and ex-covering are very similar. there
are two major differences:
(i) a-decompositions ignore vertices not in V o:(G), and
(ii) ex-coverings, while they do consist of hereditary subgraphs, do not
necessarily consist of minimal ones, as do ex-decompositions.
These differences are exemplified by the following simple ex.:rnple,
showing a graph ({a)), its ex-decomposition ((b)), and two ex-coverings ((c) and (d))
by edge-critical subgraphs:
(a..)
1 w
«..
j • • (C)
:i.. ~
• w
. . ~ (b)
• w
~ :;:. (d)
Note that the ve.:-tex x, which is not in Va does not appear in the ex.-
de~omposition. Note also that the edge {w, x} in (c), and the triangle in (d),
though they are hereditary subgraphs, are not minimal ones.
44
8.1 Every graph has an a.-covering whose elements are in A.
Proof: Let G be a graph. If G is edge-critical then the desired result holds
trivially. If G is not edge-critical, then, as in the proof of 5.2, the removal of
edges without increasing oJG) until this can no longer be done yields a subgraph
of G whose components, which are each edge-critical, constitute an a-covering
of G. D
8.::. A connected edge-critical graph G has no a-covering other than the
trivial one, {G}.
Proof: Since, for an edge-critical graph G Va(G) = V(G), any a-covering of G is
also an a-decomposition of G. In 5.1, we have shown that a connected edge-
critical graph has no ex-decomposition other than the trivial one. The desired I
( result is immediate. D
8.1 and 8.2 show, respectively, the sufficiency and the necessity of the
connected edge-critical graphs for the task of providing coverings of graphs
which respect vertex independence in the sense of our a.-coverings. It is
unfortunate indeed that this class is so large, and that so little is known about
it. In light of this, one might hope that for a given graph, it would be possible
to place some limits on which connected edge-critical graphs are needed. The
first theorem along these lines, a celebrated theorem of Konig (see [BJ, [BMJ,
[H2], or [GWJ) well predates any consideration of edge-critical graphs or of
coverings in general which respect vertex independence in the sense of our a.-
coverings.
( We shall state Konig's theorem in a way in which it is often stated, and
45
( then restate it in terms of the notions of this work.
Definition: A graph G is said to be bipartite if there exists a partition of V(G)
into sets V 1 and V 2 so that every edge of G has one of its vertices in V 1 and
the other in V 2 •
A common version of Konig's theorem may now be stated.
Konig's Theorem: Let G be a bipartite graph without isolated vertices. Let J&
be a covering of G [in our sense of covering] whose elements are each an edge
of G. If, among all such coverings of G, 1%1 is minimum, then
a_(G) 1%!. ( '
If, in Konig's theorem, we allow vertices as well as edges to serve as the
covering subgraphs, then we may insist that the covering subgraphs be disjoint,
for a pair of edges sharing a vertes, say {u, v} and {v, w}, may be replaced in a
covering by the edge {u, v} and the vertex w.
It is a well known and elementary fact that a graph is bipartite if and
only if it contains no odd cycles.
Putting these last two observations together, and noticing that the first
of them allows us to handle graphs with isolated vertices, we may restate
Konig's theorem as
Konig's Theorem: If G contains no odd cycles then G has an a-covering by
vertices and edges;
or, as
8.3. Konig's Theorem: If G contains no subgraph in L}.u then G has an
ex-covering whose elements are in A_ 1 U i.}. 0 •
46
This is highly suggestive that there may be numerous extensions of
Konig's theorem, and indeed there are.
8.4. If G contains no subgraph in L}.k U A.k+i U ... then G has an ex-covering
whose elements are in A_ 1 U i.}..0 U ... U L}.k-l·
Proof: This is a direct consequence of 8.1. D
At this point one might be tempted to say: in light of 3.2, which
guarantees that the graphs in A2, L}.3, .•. all contain odd cycles, Konig's theorem,
as stated in 8.3, is simply a special case of 8.4 (namely the case k = 1).
But this would be begging the question, for Berge's proof of 3.2 in fact
uses Konig's theorem.
However, Berge's result (3.2) generalizes a couple of earlier results, in
particular one of Beineke, Harary, and Plummer ([BHP], p. 208) which state:
Any pair of adjacent critical edges of a graph
lies on an odd cycle.
47
By a critical edge, Beineke, Harary, and Plummer mean an edge whose
removal increases a graph's independence number. Their proof of this result
does not rely on Konig's theorem, and when applied to an edge-critical graph
(where all the edges are critical) it may be stated:
Any adjacent pair of edges in an edge-critical graph
lies on an odd-cycle.
This is the same as Berge's result, but without the work "chordless," and
suffices to allow the offhand deduction of Konig's theorem (as above) without
committing the fallacy of petitio principii.
8.4, however, is a terribly weak theorem, for it essentially has infinitely
many hypotheses. Much stronger would be the following.
8.5 (Conjecture) If G contains no subgraph in .O.k then G has an ex-covering
whose elements are in .0._1 U .0.0 U ... U .O.k-1·
As is shown below, 8.5 would follow quickly from the following.
8.6 (Conjecture) If G E .D.k then G has a subgraph H E .O.k-l·
To see that 8.6 implies 8.5, suppose that 8.6 holds, and that the graph G
contains no subgraph in .O.k. So as not to contradict 8.6, G contains no
subgraph in Ak+i- By induction then, G contains no subgraph in Ak+Z• or in
.0.1'+3 , and so on ad infinitum.
(
48
Thus G contains no subgraph in l:ik U l:ik+i U ... , and by virtue of 8.4, G
must have an ex-covering whose elements are in /:i_ 1 U 6. 0 U ... U /:ik-l• which is
the conclusion of 8.5.
Such a general result as 8.6 seems at this time to be out of reach. The
next chapter is devoted to the first unresolved case in 8.5, the case k = 2.
(
49
CHAPTER 9 - Various Results.
It will be convenient to speak of coverings as including vertices and edges
of a given graph. While vertices and edges are not, strictly speaking, subsets
of a graph, this abuse of the language will not give rise to any ambiguity.
When we speak of a vertex v in this sense, let it be understood that we mean
the subgraph ({v}, ¢), and for an edge e = {u, v}, we mean the subgraph ({u, v},
{e}).
We seek a result of the following kind, for such would indeed be a
genuine extension of Konig's Theorem and a strong result:
9.1 (Conjecture) If G has no subgraph which is a very-odd-K 4 (i.e., element
of A. 2 ), then G has an a.-covering whose elements are vertices, edges, and odd
cycles (ie: elements of A.- 1 U A.0 U A. 1).
To this end we focus our attention on the following slightly more
technical conjecture from which 9.1 would follow immediately:
9.2 (Conjecture) If G E A.10 where k > 2, then G has a subgraph in A.2 •
Unfortunately, our efforts toward this end will not be entirely successful;
that is, 9 .2 will not be proved. These efforts are included, however, for a
number of reasons: they are numerous and exemplify interesting techniques;
they proceed a long way in a direction which appears to be promising; they will
allow us to prove another extension of Konig's Theorem, one which is weaker
than 9.1, and one which is already known, but whose proof relies heavily on
50
advanced theoretical notions and techniques (most notably the theory of
regular matroids). Presently, we discuss this just-mentioned extension of
Konig's Theorem, while making apparent our choice of the name very-odd-K 4 for
an element of ~2 •
In [G2], Gerards defines an odd-K 4 to be a graph of the type depicted
below:
Illustration 18:
where the wriggled lines denote pairwise vertex disjoint paths, and where the
four "faces" (triangles) of this "tetrahedron" are bound by cycles of odd length.
The graphs which we have called very-odd-K4 's comprise a proper subset
of the collection of odd-K4 's. This may be seen by sorting the odd-K/s
according to the various allowable collections of parities of the numbers of
edges in the six paths denoted by the wriggled lines. Illustration 19 depicts the
three types. The labels o (for odd) and e (for even) refer to the number of
edges in a given path. The odd-K 4 's of type I are precisely the very-odd-K 4 's,
the graphs in ~2 •
(
(
51
Illustration 19: The three types of odd-K 4 's.
Gerards continues in [G2] to prove the following extension of Konig's
Theorem, stated here in the terms to which we have become accustomed:
9.3 If G has no subgraph which is an odd-K4 then G has an ex-covering whose
elements are vertices, edges, and odd cycles.
In the next chapter, we will give a graph theoretic proof of 9.3. A subset
of that proof will provide [and this will be duly noted at the proper time] a
proof of the following more technical result.
9.4 If G E l:i.k for k > 2, then G contains a subgraph which is an odd-K4 •
[Compare this to 9.2 which conjectures: If G E Ak for k > 2, then G
contains a subgraph which is a very-odd-K 4 .]
The remainder of this chapter proceeds as follows. We will assume the
existence of a certain graph X which provides a counterexample to the
conjectured 9.2. We will also assume the existence of a certain graph Y which
provides a counterexample to 9.4. We will then let the graph Z be either of
(
52
the graphs X or Y, and show that in either case, Z (from whose supposed
existence we wish to derive a contradiction) must have some very particular
properties.
In the next chapter we will see that, if Z = Y, these properties are
sufficient to allow us to arrive at the desired contradiction, thus proving 9.4,
and, as mentioned, Gerards' extension of Konig's theorem (9.3) will follow from
this.
The following result tells us that if there exists a counterexample to 9.2
[respectively, 9.4], then there exists one which is 3-connected, Its proof relies
entirely on Gallai's characterization of the (2 but not 3)-connected edge-critical
graphs given in 3.11.
9.5 If for some k ~ 3 there exists a graph in f::.k which is (2 but not 3)
connected, and which contains no very-odd-K4 [resp. no odd-K 4], then for some j
with k 2 j 2 3, there exists a graph in f::.J which is 3-connected and contains no
very-odd-K4 [resp. no odd-K4 ].
!:roof: (Refer to 3.11) Suppose that there are edge-critical graphs with o 2 3
which are (2 but not 3)-connected and which contain no very-odd-K 4 's [resp. no
odd-K/s]. Restrict attention to such graphs of minimum o, and from among
these, let G be one with a minimum number of edges, and let o(G) = k
(i.e., G E Ak).
Let G! and G~ be as in 3.ll(b). Refer to Illustration 20, below.
Illustration 20:
I
I
\
\
I / G_,_
/
odd u-v p<:..-t-h I~ G;
\
\
.1
/
53
Suppose also, and without loss of generality, that of Gj and G2 , G~ is the
one with the property that the graph G1 obtained from it by adding the edge
{u, v} is also in A, and that G2 is the one with the property that the graph G2
obtained from it by identifying u and v is again in A.
Since k = o(G) = 6'(G 1) + o(G 2 ) - 1 (3.1 l(c)), the possibilities for the pair
(o(G 1), o(G)) are (k, 1), (k - 1, 2), (k - 2, 3), ... , (2, k - 1), (1, k).
(i) G 1 contains no very-odd-K 4 [resp. odd-K,i} for: if G1 contains such a
subgraph not using the edge {u, v}, then this subgraph would also be a
subgraph of G; and, if G1 contains such a subgraph using the edge {u, v},
then, in G, this subgraph, with {u, v} replaced by a path of odd length
from u to v lying entirely in G2 , would contradict G containing no very-
odd-K4 [odd-K4]. To see that such a path exists, let e 1 and e 2 ,
respectively, be incident with u and v in G2 . In G2, where u and v are
identified, they are adjacent, and by virtue of 3.2, they lie on a
chordless odd cycle. This odd cycle in 0 2 is the desired path of odd
length in G2 .
(ii) G2 contains no very-odd-K 4 [odd-K4 ], for: if G2 contains such a subgraph
not using the vertex which results from the identification of u and v,
then this subgraph would also be a subgraph of G; and if 0 2 contains
54
such a subgraph which does use this vertex, then, in G, this subgraph,
with u and v split, together with a path of even length in Gi connecting u
and v would again contradict G's containing no very-odd-K 4's [odd-K.1's].
The existence of such a path is guaranteed, again by virtue of 3.2, for if
e3 is an edge of G1 not equal to {u, v} but incident with u, then in 0 11 the
adjacent edges e3 and {u, v} must lie on an odd cycle. The removal of the
edge {u, v} from this cycle provides the desired path of even length from
u to v in Gf .
The minimality properties of G imply that if o(G;) > 2 then G1 will be 3-
connected, and will therefore provide the graph whose existence is asserted in
the conclusion of the theorem. If neither 6(01) > 2 nor 6(0 2) > 2, then it must
be the case that o(G1) = 6(G 2 ) = 2 in which case G1 E ~2 and G2 E ~2, and the
arguments (i) and (ii) above, ensure that G (in ~3) has a subgraph in ~2, which
is contrary to hypothesis. D
In light of 9.5, in trying to show that every connected edge-critical graph
with 6 > 2 must contain a very-odd-K 4 [odd-K4] we may restrict our attention to
those graphs which are 3-connected. That is, if there exists a counter-example
to 9.2, [or to 9.4], then there exists one which is 3-connected.
55
We now suppose the existence of counterexamples to 9.2 and 9.4, fixing
the graphs X and Y for the remainder of the chapter.
If there exists a counterexample to 9.2, let it have the following
properties (without loss of generality), and let it be called X:
(a) X is edge cri ti ca 1
(b) X is 3-connected
(c) 6(X) > 2
(d) X contains no very-odd-K 4
(e) 6(X) is minimum among all graphs with properties (a), (b), (c), and (d)
(f) e(X) is minimum among all graphs with properties (a), (b), (c), (d),
and (e).
If there exists a counterexample to 9.4 let it have the following
properties (without loss of generality), and let it be called Y:
(a) Y is edge-critical
(b) Y is 3-connected
(c) 6(Y) > 2
(d) Y contains no odd-K 4
(e) 6(Y) is minimum among all graphs with properties (a), (b), (c), and (d)
(f) e(Y) is minimum among all graphs with properties (a), (b), (c), (d),
and (e).
Let Z be either of the graphs X or Y. The propositions which follow are
true if Z = X or if Z = Y. The proofs are written as if Z = X. Where a
different argument is needed in the case Z = Y, it is given in [bracketed]
remarks.
(
56
9.6 For every vertex v of Z, p(v) ~ 3.
Proof: If some vertex v of Z has degree 2, then N(v) would comprise a cut-set
of Z of cardinality '.:, which is impossible since Z is 3-connected. 0
In [S3J, Suranyi has proved the following lemma, which we will exploit
greatly, and which is stated below in terms appropriate to this discussion.
9.7 (Suranyi, [S3J, p. 1416) If Z E Ilk for k ::e: 1, and if x is a vertex of Z,
then there exists an edge-critical subgraph H, spanning Z - x, containing no
isolated vertices and satisfying oc(H) = oc(Z) ( = a.(Z - x)).
The subgraph H is not unique. In our discussion we will let any
realization of this subgraph be denoted Z - x.
Observe that
o(Z - x) = v(Z - x) - 2a.(Z - x) = v(Z) - 1 - 2a.(Z) = o(Z) - 1.
9.8 The components of Z - x are edges and odd cycles.
Proof: Since Z - x has no isolated vertices, its components lie in
tl 0 U fl 1 U ... U fl 0czi-i· These components contain no very-odd-K4's, [odd-K/s],
for if one did, so also would Z, and thus by the minimality conditions, (e) and
(f) on Z must lie in !l0 U fl 1• D
o(Z - x) is equal to the number of odd-cycles in Z - x because of the
57
additive nature of 8 on the components of a graph, and so Z - x is a collection
of precisely o(Z) - 1 disjoint odd cycles, and perhaps some isolated edges.
As mentioned earlier, we will proceed to show that, for a given vertex x
of Z, and a given realization of the graph Z - x, the graph Z - x must satisfy
certain properties. Many of the arguments in the remainder of this chapter
take the following form: A certa.in covering of Z - x by precisely o(Z) - 1
disjoint odd cycles and the appropriate number of isolated edges cannot exist,
for if, as it did, it would provide a realization of possessing a
property which we have shown, it must possess.
9.9 The neighbors of x in Z lie on the odd cycles of Z - x.
Proof: Suppose that, on the contrary, {x, y} is an edge of Z and that {y, w} is
one of the isolated edges of Z - x. In Z, y has yet another neighbor, say z,
since p(y) :?: 3.
Illustration 21:
- -.... / '-
/ " I _j w \
I \
' ' r-/1/-' ....
' \ '9 .z:
/ \ I
' ' / ....... /
/
l.
58
a.CZ - {y, z}) > a.(Z) since Z is edge-critical. But an o:-set I of Z - {y, z}
is necessarily independent in Z - x for containing both y and z, it cannot
contain x. Thus I is independent in Z - x and of cardinality
o:(Z - {y, z}) > a.(Z) = o:(Z-x), which is absurd. D
9.10 If y is a neighbor of x then p(y) = 3, y having no neighbors other than x
and its two immediate neighbors on that odd-cycle.
Proof: Suppose, as is depicted in Illustration 22, that some neighbor y of x has
as a neighbor w, not an immediate neighbor on the cycle of Z - x to which y
belongs. Two cases are pictured. No distinction is needed in the proof.
Illustration 22:
.w -(w)
-- -
Just as in the last paragraph of the proof of 9.9, an ex-set of Z - {y, w}
would be independent in Z - x and of cardinality greater than cx(Z - x), which is
impossible. 0
9.11 For every vertex v of Z, p(v) 3.
59
Proof: Let v be a vertex of Z and let x be a neighbor of v. By 9.9 and 9.10,
in any subgraph of Z - x which qualifies for being called Z - x, v lies on an
odd-cycle of Z - x and has no neighbors other than x and its two immediate
neighbors on this cycle. 0
Proposition 9.11 implies that Z is 3-regular (!), a very strong property.
9.12 o(Z) is even and consequently greater than or equal to 4.
Proof: The sum of the degrees of the vertices of a graph must be even (being
twice the number of edges - a well known elementary result). Since in Z all
the degrees are 3, 3v(Z) must be even, requiring that v(Z) be even and
consequently that o(Z) = v(Z) - 2a.(Z) be even. D
9.13 If x has two neighbors y and z on the same odd cycle of Z - x, then on
this cycle, y and z have a common neighbor.
Proof: Suppose the contrary, with x, y, and z as above.
Illustration 23:
Ii ew 1°':;)0/o.. +ed .e~e.s
60
Then y and z divide this cycle into two y - z paths, one containing an
odd number of edges, the other containing an even number, greater than 2, of
edges. Let w be adjacent to z and on the second of these paths. The
remainder of the vertices along this path may be paired, as is shown by the
wriggled lines.
Now consider the odd cycle formed by the first of the above mentioned
paths and the edges {x, y} and {x, z}, together with the pairs of vertices just
mentioned. They together include the same number of vertices as did the
original cycle of Z - x (for x is now included, while w is omitted), and have the
same value of v - 2o:. as did that cycle, namely 1. Consequently the subgraph
consisting of this cycle and these edges has the same 6 as did the original cycle
of Z - x.
Now if, in the collection of edges and odd cycles which comprises Z - x,
we replace the original odd cycle with this new one and these isolated edges, we
61
get a graph qualified to serve as Z - w but with w having a neighbor not on
one of the odd cycles of Z - w, contrary to 9.9. 0
In particular, 9.13 implies the following two results.
9.14 Z contains no triangles.
Proof: Suppose Z contains a triangle whose vertices are x, y, and z (see
Illustration 24). y and z must lie on cycles of Z - x and since each of these
cycles must contain the edge {y, z}, they must be the same cycle.
Illustration 24:
( '
odd path
\.._ ____ ~, '- ........ __ ......
So as not to contradict 9.13, y and z must share a neighbor w on this
cycle, requiring it to be a triangle as well. 3.2 demands that in Z, the edges
{x, z} and {z, w} lie on a chordless odd cycle. This cycle, being chordless, must
necessarily a void the vertex y. This implies the existence of a pa th of an odd
number of edges from x to w, not including the vertex y. But the subgraph of
Z now depicted in Illustration 24 is indeed a very-odd-K4 [odd K4]; it is K4 with
precisely one of its edges replaced by a path of odd length. This is contrary
to our choice of Z. D
( 9.15 For no vertex x of Z and no realization of Z - x does x have all three
(
62
of its neighbors on the same odd cycle of
Proof: It is impossible for each of the 3 pairs of neighbors of x to have a
common neighbor on the same odd cycle unless that cycle is a triangle
containing only them, and this by 9.14 is impossible. D
9.16 For any vertex x of Z there exists a realization of - x for which no
edge of Z is a chord of the cycles of - x which neighbor x.
Proof: Let x be any vertex of Z, and choose any Z -x for which the sum of
the lengths of the cycles neighboring x in Z is a minimum. If there exists an
edge e of Z that is a chord of a cycle C of Z -x neighboring x in Z, then the
endpoints of e divide C into two arcs, one with an even number of edges, and
one with an odd number of edges.
Illustration 25:
-t-h~ jde.. c This even arc together with e forms an odd cycle, while the vertices
along the arc with an odd number of edges may be divided into adjacent pairs.
In Z - x, the cycle C can be replaced by this new odd cycle and the edges
formed by these adjacent pairs of edges.
(
(i) If x has a neighbor on the arc with an odd number of edges, then in
this new version of Z - x, x neighbors an isolated edge, which is
contrary to 9.9.
(ii) If x has a neighbor on the even arc, this new version of Z - x
violates the minimality condition above. 0
63
9.17 For at least one choice of the vertex x of Z, and for at least one
realization of the subgraph Z, x has as neighbors a vertex on each of
three distinct odd cycles of Z - x.
Proof: Suppose on the contrary that for every choice of the vertex x, and
every realization of Z - x, it happens that x has as neighbors two vertices on
one of the odd cycles of Z - x. Notice that 9.13 implies that these two
neighbors of x have another common neighbor on that cycle, and that 9.15
implies that the third neighbor of x be on some other odd cycle of Z - x.
The proof proceeds by induction, giving a construction which, under our
suppositions, continues ad infinitum, implying Z has an infinite number of
vertices, which is not the case.
Reference to Illustration 26 will be indispensable to the arguments which
follow. In the other illustrations in this proof, let the notation ** following
the illustration number indicate that all else, not pictured, but whose existence
has at that point been asserted, is as it is in Illustration 26.
Illustration 26:
\
\
\
\
64
c..:i..
Ci;.
65
Let x be a vertex of Z, and from among all the realizations of
which the odd cycles neighboring x are chordless in Z (see 9.16), choose one for
which the number of isolated edges is a minimum. Suppose that its connected
components are the odd cycles Ci, i = 1, 2, ... , 8(Z) - 1, and that the isolated
By hypothesis, x is adjacent to vertices u 2 and v 2 on one of the odd
cycles, call it C2 , and to u 1 on C 1• By 9.13, u 2 and v 2 have a common neighbor
w2 on C2 • Since p(w 2 ) = 3, w2 has yet another neighbor, w 3 which we will
presently show must lie on an as yet unmentioned cycle of Z - x which we will
Observe that w 3 is not on the cycle C 1, for this would imply that Z contains a
very-odd-K 4 [odd-K4 ], namely the one shown in Illustration 27, where the dashed
line denotes the arc along C1 from w3 to u 1 that has an odd number of edges.
Illustration 27: (Very-odd-K4 if w3 is on C1)
c~
66
Moreover, w3 is not on C2 , for we have taken C2 to be chordless.
Lastly, w3 is not on one of the isolated edges of Z - x, for if it were,
there would exist a realization of Z-::-w2 , for which, in Z, w2 would neighbor
one of the isolated edges (see Illustration 28). This is contrary to 9.9.
Illustration 28**: (w3 on an isolated edge of Z - x: realization of Z - w2
contradicting 9.9)
Thus w3 must lie on a previously unmentioned cycle of Z - x which we
will call C3 •
At this point we have just completed the initial step of the induction
argument. We will proceed to show that given a vertex Wzn+1' with certain
properties which will be stated presently, it must be the case that Z has
numerous previously unmentioned additional vertices, and in particular one
among them which we will call Wz 71 +3, having the very same properties which
( allowed the deduction of its existence from the existence of Wzn+i· This will
imply that Z has an infinite number of vertiees, w3 , w5 , .. ., which is absurd, and
67
when we reach this implication, the proof will be complete.
(The induction hypothesis) Let us suppose that we have shown the
existence of w2 n+ 11 being on a previously unmentioned odd cycle C2 n+i of
and being a neighbor of the previously mentioned vertex w2 ,,, lying on C2 ,,,
where W2n has neighbors U2n and V2n On C2n• and Uz 71 and V2n also have the
common neighbor x 2 n.
Observe that if we let x also be called x2 , then w3 satisfies the induction
hypothesis in the case n = 1.
For the sake of clarity, we present the induction step in terms of the
subscripts of the very next iteration, that is, the deduction of the existence of
w5 , satisfying the induction hypothesis, from the existence of w3 • The very
same arguments, with the appropriate subscripts, will allow the deduction of the
existence of w2,,+3 from that of w2n+i·
Continuing, w3 has two neighbors u 3 and v 3 on C3 • By hypothesis, in any
realization of Z - w 3 , two of the neighbors in Z of w3 have yet another common
neighbor. w2 cannot be one of these two, for neither of w2's neighbors (u2 and
v 2 ) neighbor u 3 or v 3 • So u 3 and v 3 have a previously unmentioned vertex x3
as a neighbor.
Now x3 must lie in some component of Z - x. Since each vertex has in Z
degree 3, x3 can not lie on C1 or C2 • Neither can x3 lie on C3 , for if it did, it
would have to be adjacent to u 3 (or v 3 ) along C3 , giving rise to a realization of
Z - w 2 in which in Z, w 2 neighbors one of the isolated edges of
68
Z - w2 , which cannot happen. This realization of Z - w2 is depicted in
Illustration 29.
Illustration 29**: (x3 on C3 : realization of Z - w2 contradicting 9.9)
C..z. u.%.
Vi..
\ \
\
1C::. /)(._z \ . 'X-3 U.3
WJ c.3
Also, this component of Z - x in which x3 lies cannot be yet another cycle
of Z - x, for two of the neighbors of x 3 already lie on C3 • Thus x 3 must lie on
one of the isolated edges, say {x3 , x 4}, of Z - x. ·
Consider the realization of Z - x4 depicted in Illustration 30.
69
Illustration 30**: (A realization of Z - x 4 )
In this version of Z - x 4 , two of the neighbors of x4 must share yet
another common neighbor. If one of these two is x3 , then the other must be y 3
or z 3 • But z 3 (and similarly y 3 ) neighboring x4 would provide a realization of
Z - x contradicting the condition that our original Z - x has a minimum number
of isolated edges. This realization of Z - x is shown in Illustration 31.
(
(
70
Illustration 31 **: (x 4 neighboring z 3 (or y 3): contradictory realization of Z - x)
c.3
Note that the edge {v 3, z 3} being in Z a chord of the pictured "new" C3 ,
causes no difficulty, for this is not one of the cycles neighboring x.
So x 4 has two "new" vertices as neighbors, call them u 4 and v 4 , and these
two have yet another common neighbor, call it w4 •
If we consider the realization of Z - x 4 depicted in Illustration 30, we see
that the vertices u 4 , w'l, and v 4 must lie on some previously unmentioned cycle
of Z - x, call it C4 •
Furthermore, w4 must have yet another neighbor, call it w5 • Now, w5 is
not on C1, C2 , or C3, since this would imply the existence of a subgraph of Z
which is a very-odd-K 4 [odd-K 4 ], (just as w3 could not be on C1), as shown in
Illustration 32.
71
Illustration 32: (Very-odd-K 4 if w5 is on C 1, C2, or C3 )
Nor can w5 be on C4, for this would imply the existence of a realization
of Z - x .. 11 shown in Illustration 33, for which in Z, x4 neighbors one of the
isolated edges (compare with the proof of 9.13 and Illustration 23).
Illustration 33**: (w 5 on C 4 : realization of Z - x 4 contradicting 9.9)
c'l.
\
w~/ Finally, w5 cannot be on one of the isolated edges of Z - x (as w3 could
not have been), for there would then exist a realization of Z - w4 , as depicted
in Ill ustra ti on 34 in which in Z, w 4 neighbors one of the isolated edges,
contrary to 9.9.
(
73
Illustration 34**: (w5 on an isolated edge of Z - x: realization of Z - W 4
contradicting 9.9)
C::i..
....
Thus w5 lies on a previously unmentioned· odd cycle of Z - x, call it C5 •
The vertex w5 satisfies the induction hypothesis, as did w3 • This implies
the existence of w 7 , w8 , ... , which is absurd. The proof is complete. 0
74
By virtue of the preceding remarks, we are finally at liberty to choose a
vertex x of Z, and a realization of Z - x, as depicted in Illustration 35, with
the property that in Z, x neighbors three distinct odd cycles of Z - x.
Illustration 35: (with possibly isolated edges and additional odd cycles).
CL_ ·~ \
(
75
CHAPTER 10 - Gerards' Extension of Konig's Theorem.
We are now in a position to give an elementary proof of Gerards'
extension of Konig's Theorem stated in 9.3 and repeated below:
(9.3) If G has no subgraph which is an odd-K,, then G has an O'_-covering
whose elements are vertices, edges, and odd cycles.
Proofs [of 9.3 and 9.4]: Suppose, on the contrary, there exist graphs which
contain no odd-K 4 but which do not have a-coverings by vertices, edges, and
odd cycles. Let G be one with a minimum number of edges.
As noted earlier, if {Hi> H2, •.• , Hk} is any collection of subgraphs of G
k with V(G) = U V(H;) then
!=l
k
a(G) s 2)x.CHi) i=l
since each a-set of G induces an independent set in each Hi·
If, for some edge e of G, a(G - e) = a_(G), then the graph G - e would
contain no odd-K 4 , would have no ex-covering by vertices, edges, and odd cycles
(since such would be an a-covering of G), and would have fewer edges than G.
This is contrary to our choice of G, and thus, for every edge e of G,
a.(G - e) > ex.CG). That is: G is edge-critical.
The graphs in A_ 1, A0 , and A 1 clearly have a-coverings of the desired
kind, and the graphs in A2 being very-odd-K 4's contain odd-K 4's. So 6(G) > 2.
76
It follows that there exists a connected edge-critical graph with 6 > 2, and
containing no odd-K 4 • [The contradiction which arises from the existence of
such a graph will not only complete this proof but also provide a proof of 9.4,
mentioned earlier, and again later in this chapter.]
In 9.5 it was shown that if such a graph exists, then one exists which is
3-connected.
It now follows that there exists a graph satisfying the conditions defining
the graph Y of Chapter 9, and in Chapter 9 (see 9.17) it was shown that such a
graph necessarily contains three disjoint odd cycles, CH C2 , and C3 , and a
vertex x adjacent to each of these three cycles, as shown.
( Illustration 36:
C.;. C3
) ~/
One version of Menger's Theorem ([MJ, see [BJ), a well known result in
graph theory, states:
77
It is a necessary and sufficient condition for a graph to
be k-connected that between any two vertices there exist k
disjoint paths.
It is claimed that (possible after a renaming of the cycles) there exist
such paths as are represented by the dashed lines from z 1 to z 2 and from z2 to
z 3 in lllustration 37.
Menger's theorem guarantees the existence of three x - y 2 paths. Clearly
one of these paths is the edge {x, y 2}. We may assume without loss of
generality that another of these paths avoids one of the cycles C1 and C3 , say
C3 • If the third of them avoids the cycle C1 then the claim is verified; if it
does not, then it visits C 1 before (proceeding from y 2 to x) it visits C3 • In this
case, a renaming of the cycles where C1 is renamed C2 verifies the claim.
Illustration 37:
"- LC ,,, 3 I J '2. " / ' / ' / " / '\. I /
'-.,. I /
'f'-x.
The vertices where these paths meet C2, together with x's neighbor on C2,
divide C2 into three arcs. There are four cases to be considered, according to
78
the parities of the numbers of edges in each arc.
Case 1: The three arcs are odd. Here, we immediately find an odd-K 4 of type
I, (a very-odd-K 4 ), shown below, which is a contradiction.
Illustration 38:
- ,, - - - ...._
I
'
~3 ...... ,
\
I' /
\ \
I
In Illustration 38, the paths from 2 1 to y 1 and from 2 3 to y 3 are arcs along
C1 and C3 , respectively, chosen so that the 2 2 - 2 1 - y 1 - x and 22 - z 3 - y 3 - x
paths each have an odd number of edges.
Case 2: The arc from z 2 to 22 is odd, the other two even.
(
Illustration 39: ~,
I
\
~ I
...,,. - - - .........
odd
-r.. 3 -.,
/
\
\ I
I
79
In this case, we get an odd-K4 of type II (shown below) by choosing the
arcs along C1 and C3 so as to make the z 2 - z 1 - y 1 - x and z~ - z 3 - y 3 - x
paths even. This again is a contradiction.
Illustration 40:
Cases 3 and 4: The arc from y 2 to z 2 (and by a symmetrical argument, the arc
from y 2 to z9 is odd, the other two even.
80
Illustration 41:
~I ever; 3 ~ ,,,.- - - - - -.,
I ~ z;\ \
I \
odd evej l
I \ ,, ' /
~) _j3
In this case, the arcs along C1 and C3 are chosen so as to make the
z 2 - z 1 - y 1 - x path odd and the z2 - z 3 - y 3 - x path even, yielding the odd-K4
of type III shown below, and also providing the contradiction which completes
the proof.
Illustration 42:
D
To close this chapter, we recount what has indeed been shown in our
attempt to prove conjecture 9.2 (and consequently, conjec:ture 9.1).
81
(9.4) If G is connected and edge-critical with o(G) > 2, then G contains as a
subgraph an odd-K 4 •
Proof: See the proof of 9.3 given earlier in this chapter. 0
10.1 If G is connected and edge-critical with o(G) = 3, then G contains as a
subgraph a very-odd-K4 (an element of Ll 2). [This is the first unsettled
instance of conjectures 9.2 and 8.6.]
Proof: It is a direct consequence of 9.12, which, together with the development
preceding it implies that if G is connected and edge-critical, not containing a
very-odd-K 4, and of minimum o among such graphs with o > 2, then o(G) ~ 4 (and
even). D
10.2 If G is connected and contains no very-odd-K4 and v(G) - 2cx.(G) ~ 3, then
G has an cx.-covering by vertices, edges, and odd cycles.
Proof: Suppose not, and let G be a counterexample with a minimum number of
edges. Then (as in the proof of 9.3) G is edge-critical. If o(G) < 2, G is an
isolated vertex, an isolated edge, or an odd cycle, a contradiction in any case.
If o(G) = 2, G is a very-odd-K4 , again a contradiction, while if o(G) = 3, G
contains a very-odd-K 4 by virtue of 10.2 which is once again a contradiction. 0
CH 1.PTER 11 - A note on a theorem of Graver and Yackel
A tree is a connected graph containing no cycles.
Graver ([G3J) has suggested the following notions:
Let G = (V, E) be a tree. Define a function hG with domain V U E and
codomain {O, 1} as follows:
a) for v E V,
b) for e = {u, v} E E,
if v E I for every ex-set I of G
otherwise;
if {u, v} n I rf- 0 for every cx.-set I of G, while hG(u)
otherwise.
0
82
It is this very notion, that of the function hG, which has proved to be
the seed from which grew all of the work in this thesis.
Note also that, since a tree has no cycles, a preferred ex-decomposition
(Chapter 5) of a tree contains only vertices and edges (elements of LL 1 U A0 ).
A moment's reflection reveals that the preferred ex-decomposition of G is
in fact unique, and consists of precisely those edges and vertices for which the
value of hG is 1, i.e., the support of hG.
83
The function hG may also be defined as follows:
(i) for v E V, hG(v) = cx(G) - cx(G - v), and
(ij) for e = {u, v} E E,
That the two definitions of hG agree on the vertices of G is clear. To see that
they also coincide on the edges of G, observe that hG(e), as originally defined
(1) equalling 1 implies cx(G) - cx(G - u - v) = 1 while hG(u) = hG(v) = O;
(2) equalling 0 implies either
(a) some a.-set of G misses both u and v, whence
cx(G) - a.CG - u - v) = 0 and hG(u) = hG(v) = 0, or
(b) every o:.-set of G includes u (similarly, v), in which case
cx(G) - cx(G - u - v) = 1 while hG(u) = 1 and hc(v) = 0.
In [GY] (p. 137), Graver and Yackel have shown that for a tree G = (V, E),
where gi is the number of connected subgraphs of G containing i edges.
Our second way of defining the function hG allows us to prove the
following corollary to this theorem of Graver and Yackel, and with this
proposition and its proof, we end this work.
(
84
11.1 Let G = (V, E) be a tree. For a vertex v of G (respectively, an edge e
of G) let g 1(v) (respectively, g1(e)) denote the number of connected subgraphs of
G c0ntaining the vertex v (respectively, the edge e), and having i edges. Then
IEI . hG(v) = 2:::C-U'g,(v)
i=O
and
i=l
Proof: Let v be a vertex of G. By the above remarks
hG(v) = a_(G) - cx(G - v)
where E' = E(G - v) and g~(v) is the number of connected subgraphs of G - v
containing i edges.
A connected subgraph of G not containing the vertex v contributes 0 to
the above difference, since it contributes equally to each of the two
summations. On the other hand, connected subgraphs of G co1itaining v are not
even considered in the second summation.
Thus the above difference is equal to
IEI . 2.::(-1/g;(v) as desired. i=O
85
Furthermore,
where E' = E(G - u - v), and g~ is the number of connected subgraphs of
G - u - v containing i edges.
Now a connected subgraph of G containing i edges but neither u nor v
contributes zero to the above, being counted equally in the first two
summations, and not at all in the last two.
Also, a connected subgraph of G containing i edges and u but not v
(similarly v but not u) contributes zero, being counted equally in the first and
third summations (or first and fourth) and not at all in the other two.
The only remaining connected subgraphs of G are those containing i edges
and both u and v. But such must contain the edge e for, G having no cycles,
the only path from u to v is along that edge. Also, such a subgraph
contributes equally to the first, third, and fourth summations, and not at all to
the second. Consequently, the above equals
!El . - I:C-lYgi(e),
i=l
86
the summation now beginning at i = 1 since there are no connected subgraphs of
G containing u and v and no edges. This is as desired. 0
(
BIBLIOGRAPHY
[A] B. Andrasfai: "On line-critical graphs", Theory of Graphs International
Symposium, Rome, 1966; Dunod, Paris-Gordon, and Breach, New York,
1967, 9-19.
[B] C. Berge: "Graphs and Hypergraphs'', North-Holland, Amsterdam
London, American Elsevier, New York, 1970.
[BHP] L.W. Bein eke, F. Harary, and M. Plummer: "On the critical lines of a
graph", Pacific Journal of Mathematics, 22, no. 2, (1967), 205-212.
[BM] J .A. Bondy and U .S.R. Murty: "Graph Theory with Applications",
American Elsevier, New York, 1976.
[Gl] A. George: "On line-critical graphs", (Master's thesis), Vanderbilt
University, August 1971.
[G2J A.M.H. Gerards: "An extension of Konig's theorem to graphs with no
odd-K/', (preprint).
(G3] J.E. Graver: personal communication.
[GW] J.E. Graver and M.E. Watkins: "Combinatorics with emphasis on the
theory of graphs", Springer-Verlag, New York, 1977.
87
88
[GY] J.E. Graver and J. Yackel: "Some graph theoretic results associated
with Ramsey's theorem", Journal of Combinatorial Theory, vol. 4, no. 2,
(1968), 125-175.
[Hl] A. Hajnal: "A theorem on k-saturated graphs", Canadian Journal of
Mathematics, 7, (1965), 720 - 724.
[H2] F. Harary: "Graph Theory'', Addison-Wesley, Reading, Mass., 1969.
[Kl D. Konig: "Theorie der endlichen und unendlichen Graphen", Chelsea,
New York, 1950.
[Ll] L. Lovasz: "Some finite basis theorems in graph theory", Colloquia
Mathematica Societatis Janos Bolyai 18, Combinatorics, Keszthely,
Hungary, (1976), 717 - 729.
[L2] L. Lovasz: "Combinatorial Problems and Exercises'', North-Holland,
New York, 1979.
[M] K. Menger: "Zur allgemeinen Kurventheorie", Fund, Math., 10, (1926),
96-115.
[Sl] L. Suranyi: "A note concerning a conjecture of Gallai concerning
a-critical graphs", Colloquia Mathematica Societatis Janos Bolyai 18,
Combinatorics, Keszthely, Hungary, (1976), 1065-1074.
[S2J L. Suranyi: "On a generalization of line-critical graphs", Discrete
Mathematics 30, (1980), 277-287.
[S3J L. Suranyi: "On line-critical graphs", Colloquia Mathematica Societatis
Janos Bolyai 10, Infinite and Finite Sets, Keszthely, Hungary, (1973),
1411-1444.
[W] M.E. Watkins: "On the existence of certain disjoint arcs in graphs",
Duke Mathematical Journal, vol. 35, no. 2, (1968), 231-246.
[ZJ A.A. Zykov: "On some properties of linear complexes", (Russian)
Mathematiceskii Sbornik N.S. 24(66), (1949), 163 - 188.
89
90
INDEX
adjacent ............................. 5 incident ................................. 5 o:.(G) ................................... 9 independence number .............. 9 a-covering ....................... 42 independent ........................... 8 a.-decomposi ti on ................ 20 independent set ...................... 9 ex-set ................................. 9 isolated vertex .................... 11 a.-support ......................... 20
K,, ......................................... 6 k-connected ......................... 14
bipartite graph ................. 44 Konig's Theorem ............. 44, 45
C,, ...................................... 7 maximum independent set ........ 9 chord ................................. 8 circuit ............................... ? complete graph ................... 6 connected ........................... 7 N(v) ...................................... 8 connectivity ..................... 14 neighbor ................................ 5 covering ........................... 42 neighbor set .......................... 8 cube ................................ 26 cut-set ............................. 11 cycle ................................. 7 octahedron, .......................... 25
odd-K 4 ................................. 50 odd subdivision of K 4 ........... 12
deficiency .......................... 9 degree ................................ 7
D.. ..................................... 12 L'.ln ................................... 12 path ...................................... 7 o(G) ................................... 9 platonic graph ...................... 24 dodecahedron .................... 28 preferred ex-decomposition .... 20
E(G) ................................... 5 p(v) ....................................... 7 e(G) ................................... 9 edge ................................... 5 edge-critical graph ............ 10 span .................................... 11 endpoint ............................. 5 subgraph ............................... 8
finite basis theorem .......... 13 tetrahedron ......................... 24 tree ..................................... 82
G - e ................................. 8 G - S ................................. 8 V(G) ...................................... 5 G - v ................................. 8 v(G) ...................................... 9 graph ................................. 5 V a(G) .................................. 20
valence .................................. 7 vertex ................................... 5
hG .................................... 82 very-odd-K 4 ......................... 12 hereditary ........................ 18
Z - x ................................... 56 icosahedron ...................... 26
BIOGRAPHICAL DAT A
VINCENT EDWARD FA TICA
Date and Place of Birth
21 February 1949
Albany, New York
Elementary School
Saint Patrick's School
Ravena, New York
High School
The Vincentian Institute
Albany, New York
Higher Ed uca ti on
B.S., Syracuse University 1972
M.S., Syracuse University 1977
M.Ph., Syracuse University 1985
91
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