Enumeration of Unrooted Odd-Valent Regular Planar Maps

Abstract

We derive closed formulae for the numbers of rooted maps with a fixed number of vertices of the same odd degree except for the root vertex and one other exceptional vertex of degree 1. The same applies to the generating functions for these numbers. Similar results, but without the vertex of degree 1, were obtained by the first author and Rahman. We also show, by manipulating a recursion of Bouttier, Di Francesco and Guitter, that there are closed formulae when the exceptional vertex has arbitrary degree. We combine these formulae with results of the second author to count unrooted regular maps of odd degree. In this way we obtain, for each even n, a closed formula for the function f n whose value at odd positive integers r is the number of unrooted maps (up to orientation-preserving homeomorphisms) with n vertices and degree r. The formula for f n becomes more cumbersome as n increases, but for n > 2 each has a bounded number of terms independent of r.