By C.Paul Bonnington

This isn't a conventional paintings on topological graph concept. No present graph or voltage graph ornaments its pages. Its readers won't compute the genus (orientable or non-orientable) of a unmarried non-planar graph. Their muscular tissues won't flex less than the tension of lifting walks from base graphs to derived graphs. what's it, then? it truly is an try to position topological graph concept on a simply combinatorial but rigorous footing. The car selected for this function is the con­ cept of a 3-graph, that's a combinatorial generalisation of an imbedding. those safely edge-coloured cubic graphs are used to categorise surfaces, to generalise the Jordan curve theorem, and to turn out Mac Lane's characterisation of planar graphs. hence they playa significant function during this publication, however it isn't being urged that they're inevitably the best device in components of topological graph concept now not handled during this quantity. Fruitful notwithstanding 3-graphs were for our investigations, different jewels has to be tested with a special lens. the only real requirement for realizing the logical improvement during this publication is a few straightforward wisdom of vector areas over the sector Z2 of residue sessions modulo 2. teams are sometimes pointed out, yet no services in crew concept is needed. The remedy can be liked top, although, through readers accustomed to topology. A modicum of topology is needed for you to understand a lot of the incentive we stock for the various recommendations introduced.

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A' .. ",. • ~.................. ~...... • _ ..... 9. I ;; ••• • Y d) ... I ... "" \... :, \..... ". : \ ..... ',. ~--(' Z W I : • •.... -- I r. ,' . " . r - •• ..... --....... : .... ,,' . • ..... ' I b) I ............ --,..... ". , I 47 ............. '. • ................ - ......... 48 3. Classification of Surfaces The above process transforms U into another unitary gem U'. The handle {X, Y} has been replaced by the assembled handle {W, Z}. The red edges in any assembled cross-cap or in a member of any other assembled handle are in These cross-caps and handles therefore remain assembled.

31 -n~ '. 8. It may also be true that the red-blue bigons of a 3-graph are all squares. In this case, the 3-graph is called a gem. ] The red-blue bigons of a gem sometimes are called bisquares. Let K be a gem (H, V, F). Every red-blue bigon has two red edges, each of which belongs to a unique red-yellow bigon. Thus, each bisquare meets just two red-yellow bigons, which may not be distinct. Let G(K) be the graph in which VG(K) is the set of red-yellow bigons, EG(K) is the set of red-blue bigons, and each edge joins the vertices it meets.

Thus, non-empty connected planar graphs are the graphs imbeddable in the sphere. They are the ones that underlie spherical gems. In general, the determination of the genus of a graph is an unsolved problem. However, a number of characterisations of planar graphs are known, and we describe several in this book. The reader is referred to Gross and Tucker (1987) for an account of techniques used to find the genus of a graph. 3 and the fact that IV K I = 4IY(K)1 for any gem K. 9 Let K be a connected gem.

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