Basic graph-theoretic terms

In this section we shall define some basic graph-theoretic terms that will be used in this book. Other graph-theoretic terms which are not included in this section will be defined when they are needed.

Unless stated otherwise, all graphs are finite, undirected, simple and loopless. A directed graph is called a digraph and a directed edge is called an arc. A multigraph permits more than one edge joining two of its vertices. The number of edges joining two vertices u and v is called the multiplicity of uv and is denoted by μ(u,v).

The cardinality of a set S is denoted by |s|. Let G = (V,E) be a graph where V = V(G) is its vertex set and E = E(G) is its edge set. The order (resp. size) of G is |v| (resp. |E|) and is denoted by |G| (resp. e(G)). Two vertices u and v of G are said to be adjacent if uv ε E. If e = uv ε E, then we say that u and v are the end-vertices of e and that the edge e is incident with u and v. Two edges e and f of G are said to be adjacent if they have one common end-vertex. If uv ε E, then we say that v is a neighbour of u. The set of all neighbours of u is called the neighbourhood of u and is denoted by NG(U) or simply by N(u) if there is no danger of confusion. The valency (or degree) of a vertex u is |N(u)| and is denoted by d(u).

A complete, optional course on Graph Theory was first offered to Fourth Year Honours students of the Department of Mathematics, National University of Singapore in the academic year 1982/83. To those students taking this course, it was their first introduction to Graph Theory and so the standard of the course could not be set too high. However, since it was a fourth year Honours Course, the standard could not be too low. For this reason, I decided to use some existing textbooks for the basic results in the first term and to concentrate on only a few special topics in the second term in order to expose the students to some very recent results. This book eventually grew out from the lectures I gave to the students during the academic years 1982/83 and 1983/84.

More than seventy per cent of the materials in this book are taken directly from recent research papers. Each chapter (except chapter 1) gives an up-to-date account of a particular topic in Graph Theory which is very active in current research. In addition, detail of proofs of all the theorems are given and numerous exercises and open problems are included. Thus this book is not only suitable for use as a supplement to a course text at advanced undergraduate or postgraduate level, but will also, I hope, be of some help to researchers in Graph Theory. In fact, Mr. Chen Jing-Hui had written to inform me that by using my lecture notes in his fourth year Graph Theory course in Xiamen University, his students were able to do some research straightway.

The automorphism group of a graph

Suppose G and H are two graphs. If Φ : V(G) → V(H) is an injection such that xy ε E(G) implies that Φ(x)Φ(y) ε E(H) for any edge xy of G, then Φ is called a monomorphism from G to H. If there exists a monomorphism from G to H, then we say that G is embeddable in H. Two graphs G and H are isomorphic if there is a bijection Φ : V(G) → V(H) such that xy ε E(G) if and only if Φ(Kx)Φ(y)ε E(H). An isomorphism from G onto itself is called an automorphism of G. The set of all automorphisms of G forms a group under the composition of maps and is denoted by Aut G, A(G) or simply by A, Thus we can consider A(G) as a group acting on V(G) which preserves adjacency. The automorphism group A(G) of G measures the degree of symmetry of G. If A(G) is the identity group, then G is called an asymmetric graph.

It was known to Riddell [51] that for large integers p, almost all labelled graphs of order p are asymmetric. Erdös and Rényi [63] gave a proof of this result using probabilistic methods. An outline of a proof of this result can also be found in Harary and Palmer [73; p. 206]. Wright [71,74] proved the same result for unlabelled graphs and Bollobás [82] generalized Wright's result to unlabelled regular graphs.

Suppose G is a graph. If G is asymmetric, then any vertex of G is distinguishable from all its other vertices.

Introduction and definitions

The notion of an edge-colouring of a graph can be traced back to 1880 when Tait tried to prove the Four Colour Conjecture. (A detailed account of this can be found in many existing text books on Graph Theory and therefore we shall not repeat it here.) However, there was not much development during the period 1881–1963, A breakthrough came in 1964 when Vizing proved that every graph G having maximum valency Δ can be properly edge-coloured with at most Δ + 1 colours (“proper” means that no two adjacent edges of G receive the same colour). This result generalizes an earlier statement of Johnson [63] that the edges of every cubic graph can be properly coloured with four colours.

Many of the results of this chapter will be concerned with the socalled ‘critical graphs’ introduced by Vizing in the study of classifying which graphs G are such that x'(G) = Δ(G) + 1. The main reference of this chapter is Fiorini and Wilson [77].

We now give a few definitions. Let G be a graph or multigraph. A (proper, edge-) colouring π of G is a map π : E(G) → {1,2,…} such that no two adjacent edges of G have the same image. The chromatic index χ'(G) of G is the minimum cardinality of all possible images of colourings of G. Hence, if Δ = Δ(G), then it is clear that χ'(G) > Δ and Vizing's theorem says that Δ < χ'(G) < Δ + 1. If χ'(G) = Δ, G is said to be of class 1, otherwise G is said to be of class 2.

Introduction and definitions

We can instal a graph G of order n into a computer by encoding the entries of the upper triangular part of its adjacency matrix. One problem arises naturally : “Can we find, in the worst case, whether the graph G has a specific property P, without decoding all the n(n–1)/2 entries of the upper triangular part of its adjacency matrix?”

The main objective of this chapter is to introduce a Two Person Game to tackle the above problem.

Let Gn be the set of all graphs of order n and let F ⊆. Gn be the set of all graphs such that each of its members has property P. To see that whether a graph G (of order n) possesses property P or not, it is equivalent of showing whether G belongs to F or not. Hence we can introduce the Two Person Game in a general setting and treat the graph property as a special case.

Let T be a finite set of cardinality |T| = t and let p(T) be the power set of T, i.e. the set of all subsets of T. We call F ⊆ p(T) a property of T. A measure of the minimum amount of information necessary, in the worst case, to determine membership of F is as follows. Suppose two players, called the Constructor (Hider) and Algy (Seeker), play the following game which we also denote by F. Algy asks questions of the Constructor about a hypothetical set H ⊂ T.

Introduction and definitions

Suppose G1, G2, …, Gκ are graphs of order at most n. We say that there is a packing of G1, G2, …, Gκ into the complete graph κn if there exist injections αi : V(Gi) + V (κn), i = 1,2,…, κ such that α*i(E(Gi)) ∩ α*j(E(Gj)) = Φ for i ≠ j, where the map α*i : E(Gi) → E(kn) is induced by αi. Similarly, suppose G is a graph of order m and H is a graph of order n ≥ m and there exists an injection α : V(G) → V(H) such that α*(E(G)) ∩ E(H) = Φ. Then we say that there is a packing of G into H, and in case n = m, we also say that there is a packing of G and H or G and H are packable. Thus G can be packed into H if and only if G is embeddable in the complement H of H. However, there is a slight difference between embedding and packing. In the study of embedding of a graph into another graph, usually at least one of the graphs is fixed whereas in the study of packing of two graphs very often both the graphs are arbitrarily chosen from certain classes.

In practice, one would like to find an efficient algorithm to pack two graphs G and H. But this has been shown to be an NP-hard problem (see Garey and Johnson [79;p.64]).