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

Short Solution of Kotzig's Problem for Bipartite Graphs

AbstractIn 1975, A. Kotzig posed the following problem: LetGbe at-regular graph which has a proper edget-coloring,t⩾4. Is it possible to obtain, from one proper edget-coloring ofG, any other proper edge t-coloring of G using only transformations of 2-colored and 3-colored subgraphs such that the intermediate colorings are also proper? The author and A. N. Mirumian proved that it is possible ifGis a bipartite graph. We give here a short proof of this result