Edge-colouring seven-regular planar graphs

TL;DR

This paper provides a simplified proof for a conjecture on edge-coloring seven-regular planar multigraphs, extending known results for degrees up to 8 and relating to the four-color theorem.

Contribution

It offers a simpler proof for the seven-regular case of a conjecture on edge-coloring planar multigraphs, building on previous work for degrees up to 8.

Findings

Proof of the conjecture for d=7 using simplified methods

Extension of edge-coloring results to seven-regular planar multigraphs

Connection to the four-color theorem

Abstract

A conjecture due to the fourth author states that every $d$-regular planar multigraph can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and its complement. For $d = 3$ this is the four-colour theorem, and the conjecture has been proved for all $d\le 8$, by various authors. In particular, two of us proved it when $d=7$; and then three of us proved it when $d=8$. The methods used for the latter give a proof in the $d=7$ case that is simpler than the original, and we present it here.