  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.
