Enumeration of labelled 4-regular planar graphs

TL;DR

This paper introduces a recursive combinatorial method to count labelled 4-regular planar graphs, enabling effective computation of their generating functions and enumeration of related graph classes.

Contribution

It provides the first systematic recursive scheme for enumerating labelled 4-regular planar graphs and related structures, advancing combinatorial enumeration techniques.

Findings

Effective computation of exponential generating functions for labelled 4-regular planar graphs

Enumeration of labelled 3-connected 4-regular planar graphs

Enumeration of simple 4-regular rooted maps

Abstract

We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.