Gerhard Ringel

Plane Constructions for Graphs, Networks, and Maps Measurements of Planarity

Consider a mining project where four substances sand, turf, coal, and earth are being extracted at five different sources. One also has four locations where the different materials will be collected, labelled S, T, C, and E respectively.

Construction by Induction

With the help of the following theorem we will solve the non-orientable Case 1. We do not really need the orientable version of Theorem 10.1 because we already have easy solutions in Cases 1 and 7.

Orientable Case 0

Since this is a regular case we shall construct an orientable triangular embedding of K12s using a current graph of index 1. But this time the currents will come from a non-abelian group of order 12 s, and it is to be expected that some of the construction principles for a current graph have to be a little modified.

Orientable Cases 1, 4, and 9

Now we will construct a triangular embedding of K n into an orientable surface for each n of the form n= 12s+ 4.

Orientable Cases 11, 2, and 8

Let the positive integer n be of the form n= 12 s +11. In harmony with the general outline in Section 5.3 we construct a triangular embedding of K n — K5 into a closed orientable surface. This means we will find a triangular rotation of K n — K5.

Non-Orientable Cases (Index 1)

It is our intention to determine the non-orientable genus of K n which means to prove formula (4.19). For n ≦ 8 this was already done in Sections 4.6 and 5.1.

Problems, Illustrations, History

In this first chapter we will use geometric intuition and geometric imagination to explain all the problems and all the results which are presented in this book. In the first chapter there will be no proofs at all. It is hoped that this will serve as an introduction for those not familiar with the subject. The proofs and the mathematical foundations of graph theory and the theory of surfaces will be presented in later chapters.

Classification of Surfaces

In this chapter we will present material that is well known; however we need not only the results, but also parts of the proofs as we shall see in later chapters. For more details of this theory see H. Seifert and Threlfall [81] or M. Frechet and Ky Fan [22].

Combinatorics of Embeddings

In order to prove, the Map Color Theorem we only have to determine the genus and the non-orientable genus of the complete graph K n , according to Theorems 4.9 and 4.10. That means we just have to prove Eqs. (4.13) and (4.19). Denote the right hand side of (4.13) by p and that of (4.19) by q. Then we have to exhibit an embedding of K n into S p and into N q . The following table shows the values of p and q respectively for small values of n.

Graphs on Surfaces

Given a graph G and a surface S, one asks the following question. Is it possible to “draw” this graph G on the surface S such that the arcs of G intersect only at their common vertices? This concept must be precisely defined. Remember that a surface S is just a set of polyhedra. (See the definition in Section 3.6.)