On the Number of 1-Factors of Locally Finite Graphs

AbstractEvery infinite locally finite graph with exactly one 1-factor is at most 2-connected is shown. More generally a lower bound for the number of 1-factors in locally finite n-connected graphs is given

Scenic trails ascending from sea-level Nim to alpine chess

Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Ascend from Nim to chess in small strides at a gradient that's not too steep. Presentation: Informal; examples of games sampled from various strategic viewing points along scenic mountain trails, which illustrate the theory.Comment: 25 page

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