3 research outputs found
Recommended from our members
Edge-colourings of graphs
All the results in this thesis are concerned with the classification of graphs by their chromatic class.
We first extend earlier results of Fiorini and others to give a complete list of critical graphs of order at most ten. We give conditions for extending the edge-colouring of a nearly complete subgraph to the whole graph and use this result to prove a special case of Vizing's conjecture. We also use other methods to solve further cases of this conjecture.
A major part of the thesis classifies those graphs with at most 4 vertices of maximum degree and this work is generalised to graphs with r vertices of maximum degree. We also completely classify all regular graphs G with degree at least 6/7|V(G)|.
Finally we give some examples of even order critical graphs and introduce the concept of a supersnark
Recommended from our members
The chromatic index of simple graphs
The object of this thesis is twofold:
(i) to study the structural properties of graphs which are critical with respect to edge-colourings;
(ii) to apply the results obtained to the classification problem arising from Vizing's Theorem.
Chapter 1 contains a historical, non-technical introduction, general graph-theoretic definitions and notation, a discussion of Vizing's Theorem as well as a survey of the main results obtained to date in Vizing's classification problem. Chapter 2 introduces the notion of criticality in the first section; the second section contains both well-known and new constructions of critical graphs which will be used in later chapters. The third and final section contains new results concerning elementary properties of critical graphs. Chapter 3 deals with uniquely-colourable graphs and their relationship to critical graphs. Chapter 4 contains results on the connectivity of critical graphs, whereas Chapter 5 deals with bounds on the number of edges of these graphs. In particular, bounds improving those given by Vizing are presented. These results are applied to problems concerning planar graphs. In Chapter 6, critical graphs of small order are discussed. All such graphs of order at most 8 are determined, while the 'critical graph conjecture’ of Beineke & Wilson and Jakobsen is shown to be true for all graphs on at most 10 vertices. The seventh and final chapter deals with circuit length properties of critical graphs. In particular, the minimal order of certain critical graphs with given girth and maximum valency is determined. Results improving Vizing’s estimate of the circumference of critical graphs are also given. The Appendix includes a computer programme which generates critical graphs from simpler ones using a constructive algorithm given in Chapter 2