Aspects of graph colouring

The four-colour conjecture of 1852, and the total colouring conjecture of 1965, have sparked off many new concepts and conjectures. In this thesis we investigate many of the outstanding conjectures, establishing various related results, and present many conjectures of our own. We give a brief historical introduction (Chapter 1) and establish some notation, terminology and techniques (Chapter 2). Next, in Chapter 3, we examine the use of latin squares to represent edge and total colourings. In Chapters 4 - 6 we deal with vertex, edge and total colourings respectively. Various ways of measuring different aspects of graphs are presented, in particular, the ‘colouring difference’ between two edge-colourings of a graph (Chapter 5) and the ‘beta parameter’ (defined in Chapter 2 and used in Chapters 3 and 6); this is a measure of how far from a type 1 graph a type 2 graph can be. In Chapter 6 we derive an upper bound for the beta value of any near type 1 graph and give the exact results for all Kn. The number of ways of colouring Kn and Kn,,n are also quantified. Chapter 6 also examines Hilton’s concept of conformability. It is shown that every graph with at least A spines is conformable, and an extension to the concept, which we call G*-conformability, is introduced. We then give new necessary conditions for a cubic graph to be type 1 in relation to G*-conformability. Various methods of manipulating graphs are considered and we present: a method to compatibly triangulate a graph G-e; a method of introducing a fourth colour thus allowing a sequence of Kempe interchanges from any edge 3-colouring of a cubic graph to any other; and a method to re-colour a near type 1 graph within a certain bound on beta. We end this thesis with a brief discussion on possible practical uses for colouring graphs. A list of the main results and conjectures is given at the end of each chapter, but a short list of the principle theorems proven is given below

Complete Sets of Mutually Orthogonal Hypercubes and Their Connections to Affine Resolvable Designs

Recently, Laywine and Mullen proved several generalizations of Bose\u27s equivalence between the existence of complete sets of mutually orthogonal Latin squares of order n and the existence of affine planes of order n. Laywine further investigated the relationship between sets of orthogonal frequency squares and affine resolvable balanced incomplete block designs. In this paper we generalize several of Laywine\u27s results that were derived for frequency squares. We provide sufficient conditions for construction of an affine resolvable design from a complete set of mutually orthogonal Youden frequency hypercubes; we also show that, starting with a complete set of mutually equiorthogonal frequency hypercubes, an analogous construction can always be done. In addition, we give conditions under which an affine resolvable design can be converted to a complete set of mutually orthogonal Youden frequency hypercubes or a complete set of mutually equiorthogonal frequency hypercubes

Construction of Complete Sets of Mutually Equiorthogonal Frequency Hypercubes

Equiorthogonal frequency hypercubes are one particular generalization of orthogonal latin squares. It has been shown previously that a set of mutually equiorthogonal frequency hypercubes (MEFH) of order n and dimension d, using m distinct symbols, can have at most (n − 1)d/(m − 1) hypercubes. In this article, we show that this upper bound is sharp in certain cases by constructing complete sets of (n − 1)d/(m − 1) MEFH for two classes of parameters. In one of these classes, m is a prime power and n is a power of m. In the other, m = 2 and n = 4t, provided that there exists a Hadamard matrix of order 4t. In both classes, the dimension d is arbitrary. We also provide a Kronecker product construction which can be used to yield sets of MEFH in which the order is not a prime power