Enumeration of Regular Maps on Surfaces of a Given Genus

TL;DR

This paper develops methods to count rooted and unrooted d-regular maps on oriented surfaces of genus g, with special focus on maps with a single face, advancing combinatorial enumeration techniques.

Contribution

It provides new enumeration formulas for d-regular maps on surfaces of arbitrary genus, including detailed analysis of maps with a single face.

Findings

Derived enumeration formulas for rooted and unrooted d-regular maps.

Analyzed the special case of maps with a single face in detail.

Extended combinatorial enumeration to surfaces of arbitrary genus.

Abstract

In the present work the rooted and unrooted d-regular maps on 2-dimentional oriented surfaces of genus g are enumerated. Separately and in more detail the case of d-regular maps with a single face are considered.