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 vertex of degree 1. A similar result, but without the vertex of degree 1, was obtained by the first author and Rahman. These formulae are combined with results of the second author to count unrooted regular maps of odd degree. We succeed in finding, for each even n, a closed formula f n (r) for the number of unrooted maps (up to orientation-preserving homeomorphisms) with n vertices and odd degree r, provided r is an odd prime or gcd(r, n − 2) = 1 or n = 2. The functions f n become more cumbersome as n increases, but for n > 2 each has a bounded number of terms independent of r.

Modular groups and planar maps

“…A particular case consists of the regular maps having the same degree at all the vertices. For this class, closed recursive formulas have been obtained for Eulerian or even-valent maps [7,13,12,20,31,36]. However, for regular odd-valent maps, the situation is not as simple except for the case n = 3 where we have a closed formula for enumerating the trivalent rooted maps due to Mullin [20], and a formula for enumerating the trivalent unrooted maps due to Liskovets [12].…”