Non-recursive Counts of Graphs on Surfaces

TL;DR

This paper derives explicit, non-recursive formulas for counting maps with even-valent vertices embedded in surfaces of any genus, advancing combinatorial enumeration in topology and graph theory.

Contribution

It provides the first closed-form expressions for map counts with arbitrary genus and even-valent vertices, including new higher genus examples.

Findings

Explicit formulas for map counts with even-valent vertices

Closed-form expressions valid for any genus

New higher genus 4-valent map examples

Abstract

The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of the surface. As such this problem lies at the cross-roads of combinatorial studies in low dimensional topology and graph theory. The determination of explicit formulae for map counts, in terms of closed classical combinatorial functions of $g$ and $j$ as opposed to a recursive prescription, has been a long-standing problem with explicit results known only for very low values of $g$. In this paper we derive closed-form expressions for counts of maps with an arbitrary number of even-valent vertices, embedded in surfaces of arbitrary genus. In particular, we exhibit a number of higher genus examples for 4-valent maps that have not appeared prior in the literature.