Edge-colouring seven-regular planar graphs

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 $d = 3$thisisthefour-colourtheorem,andtheconjecturehasbeenprovedforall$d\le 8$,byvariousauthors\mathrm{.}Inparticular,twoofusproveditwhen$d=7$; 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.