Enumeration of 4-regular one-face maps

TL;DR

This paper provides explicit formulas for counting 4-regular maps with a single face, both labeled and unlabeled, contributing to combinatorial map enumeration.

Contribution

It introduces explicit enumeration formulas for 4-regular one-face maps, advancing combinatorial map counting methods.

Findings

Derived formulas for labeled maps

Derived formulas for unlabeled maps

Enhanced understanding of 4-regular map enumeration

Abstract

We give explicit formulas enumerating 4-regular labelled and unlabelled one-face maps.