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=3d=3 this is the four-colour theorem, and the conjecture has been proved for all d≤8d≤8, by various authors. In particular, two of us proved it when d=7d=7; and then three of us proved it when d=8d=8. The methods used for the latter give a proof in the d=7d=7 case that is simpler than the original, and we present it here.