On a conjecture by Anthony Hill

Abstract

In the 1950's, English painter Anthony Hill described drawings of complete graphs K n in the plane having preciselyIt became a conjecture that this number is minimum possible and, despite serious efforts, the conjecture is still widely open. Another way of drawing K n with the same number of crossings was found by Blažek and Koman in 1963. In this note we provide, for the first time, a very general construction of drawings attaining the same bound. Surprisingly, the proof is extremely short and may as well qualify as a "book proof". In particular, it gives a very simple explanation of the phenomenon discovered by Moon in 1968 that a random set of n points on the unit sphere S 2 in R 3 joined by geodesics gives rise to a drawing whose number of crossings asymptotically approaches the Hill value H(n).