Edge-colouring eight-regular planar graphs

TL;DR

This paper proves a longstanding conjecture that every eight-regular planar graph can be edge-colored with eight colors, extending known results for lower degrees and confirming a 1973 conjecture.

Contribution

The paper establishes the first proof for the case d=8, completing the verification for all d ≤ 8 in the conjecture about edge-coloring regular planar graphs.

Findings

Confirmed the conjecture for d=8

Extended the known range of d for which the conjecture holds

Provided new techniques for edge-coloring in planar graphs

Abstract

It was conjectured by the third author in about 1973 that every $d$-regular planar graph (possibly with parallel edges) 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 $d\le 7$, by various authors. Here we prove it for $d = 8$.