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This is not a traditional work on topological graph theory. No
current graph or voltage graph adorns its pages. Its readers will
not compute the genus (orientable or non-orientable) of a single
non-planar graph. Their muscles will not flex under the strain of
lifting walks from base graphs to derived graphs. What is it, then?
It is an attempt to place topological graph theory on a purely
combinatorial yet rigorous footing. The vehicle chosen for this
purpose is the con cept of a 3-graph, which is a combinatorial
generalisation of an imbedding. These properly edge-coloured cubic
graphs are used to classify surfaces, to generalise the Jordan
curve theorem, and to prove Mac Lane's characterisation of planar
graphs. Thus they playa central role in this book, but it is not
being suggested that they are necessarily the most effective tool
in areas of topological graph theory not dealt with in this volume.
Fruitful though 3-graphs have been for our investigations, other
jewels must be examined with a different lens. The sole requirement
for understanding the logical development in this book is some
elementary knowledge of vector spaces over the field Z2 of residue
classes modulo 2. Groups are occasionally mentioned, but no
expertise in group theory is required. The treatment will be
appreciated best, however, by readers acquainted with topology. A
modicum of topology is required in order to comprehend much of the
motivation we supply for some of the concepts introduced."
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