On a conjecture by Anthony Hill

TL;DR

This paper presents a simple, general construction of drawings of complete graphs that achieve the minimal number of crossings conjectured by Anthony Hill, providing insights into the asymptotic behavior of crossings in random spherical point sets.

Contribution

It introduces a very general, concise construction method for drawings of $K_n$ with the minimal number of crossings, supporting Hill's conjecture and explaining Moon's asymptotic findings.

Findings

Construction attains Hill's crossing bound for all n

Short proof qualifies as a 'book proof'

Explains Moon's asymptotic crossing number phenomenon

Abstract

In the 1950's, English painter Anthony Hill described drawings of complete graphs $K_n$ in the plane having precisely $$H(n) = \tfrac{1}{4}\lfloor \tfrac{n}{2}\rfloor \, \lfloor \tfrac{n-1}{2}\rfloor \, \lfloor \tfrac{n-2}{2}\rfloor \,\lfloor \tfrac{n-3}{2}\rfloor$$ crossings. It 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 Bla\v{z}ek 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 $\SS^2$ in $\RR^3$ joined by geodesics gives rise to a drawing whose number of crossings asymptotically approaches the Hill value $H(n)$.