Uniform Plane Graphs

Abstract

This paper presents two special classes of plane graphs characterized by the sequences M(v) and W(e). M(v) is the circular sequence consisting of the numbers of vertices on the meshes around vertex v ; and W(e) is the sequence consisting of the numbers of vertices on the meshes to the right and the left of edge e and also of the degrees of the head and the tail of e. A graph is called uniform with respect to M(v) or W(e) if its vertices all have the same M(v), or if its edges all have the same W(e), respectively. It is shown that if such a uniform plane graph exists for the given M(v) or W(e), the numbers of its vertices, edges and meshes are uniquely determined. Then, the conditions on M(v) or W(e) for the existence of a graph are investigated. Tables of plane graphs which are uniform with respect to M(v) or W(e) are presented. Besides regular polyhedrons, there are thirteen types of graphs which are uniform with respect to M(v), and only four graphs which are uniform with respect to W(e).