The geometry of planar graphs

Summary

Introduction

Every undergraduate course in graph theory mentions basic results about finite planar graphs – Kuratowski's criterion for embeddability, Euler's Theorem, and so on. However the corresponding results for infinite graphs seem to be little known. It turns out that the concept of embeddability in the plane has many ramifications and variants in the infinite case, and one of the purposes of this exposition is to survey these. For the most part results will only be quoted and no proofs given – for these the reader is referred to the literature listed in the bibliography.

In this survey we aim to show how fruitful is the interaction between the theories of finite and of infinite planar graphs. Results from one of these fields often inspire nontrivial problems in the other, and frequently suggest analogous questions about graphs embeddable in manifolds of arbitrary genus.

Although it may seem foreign to the subject, especially to those only interested in problems of a strictly combinatorial or topological nature, quite a large part of what we shall do will be metrical in character. There are several reasons for this. For example, in our discussion of Euler's Theorem and its variants in Section 3, the results are not true unless we impose quite strong restrictions on the kinds of graph we are considering – and we only know how to formulate these restrictions in metrical terms.