A short solution of Heawoodʹs empire problem in the plane Original Research Article

In his well-known paper of 1890 where he demolished Kempeʹs ‘proof’ for the four-colour theorem P.J. Heawood showed that the empires of every multimap (a map whose face set is partitioned into empires) can be coloured by 6r colours if r is the maximal number of faces belonging to some empire. He conjectured that 6r colours are necessary, too, for colouring every such multimap if r ⩾ 2, and proved it in case r = 2 by giving a multimap with 12 mutually neighbouring empires (complete multimap). Further, such complete multimaps were given by others in cases r = 2, 3, 4. In 1984 Jackson and Ringel proved all cases r ⩾ 5. We shall give a shorter and nearly uniform proof of Heawoodʹs conjecture, using copies of a special symmetrical graph together with an evident labelling of their vertices for combining the desired complete multimaps.