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
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Date
1991
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Massey University
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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.
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Keywords
Graph theory, Combinatorial topology