Combinatorial maps and the foundations of topological graph theory : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University

Abstract

This work.develops the foundations of topological graph theory with a unified approach using combinatorial maps. (A combinatorial map is an n-regular graph endowed with proper edge colouring in n colours.) We establish some new results and some generalisations of important theorems in topological graph theory. The classification of surfaces, the imbedding distribution of a graph, the maximum genus of a graph, and MacLane's test for graph planarity are given new treatments in terms of cubic combinatorial maps. Among our new results, we give combinatorial versions of the classical theorem of topology which states that the first Betti number of a surface is the maximum number of closed curves along which one can cut without dividing the surface up into two or more components. To conclude this thesis, we provide an introduction to the algebraic properties of combinatorial maps. The homology spaces and Euler characteristic are defined, and we show how they are related.