A short solution of Heawoodʹs empire problem on the torus and implications for higher genus Original Research Article

In his well-known paper of 1890, where he demolished Kempeʹs ‘proof’ for the four-colour theorem P.J. Heawood gave an upper bound for the number of colours necessary for colouring the empires of any multimap (a map on some orientable closed surface whose face set is partitioned into empires) depending on the genus γ of that surface and the maximal number r of faces belonging to some empire. He conjectured that this bound is sharp except possibly if γ=0,r=1 (the four-colour problem) and proved it in case γ=0,r=2. Further affirmative partial results were given in various contributions by G. Ringel and others, among them verifications for γ⩾1,r=1 and 0⩽γ⩽2,r⩾2. We shall give a shorter proof of Heawoodʹs conjecture for γ=1,r⩾1, based on an elementary construction, and its implications for higher genus.