A short solution of Heawood's empire problem in the plane

✍ Scribed by Walter Wessel

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 226 KB
• Volume: 191
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(98)00110-1

✦ Synopsis

In his well-known paper of 1890 where he demolished Kempe's '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 Heawood's conjecture, using copies of a special symmetrical graph together with an evident labelling of their vertices for combining the desired complete multimaps.