Hill’s conjecture for the crossing number of complete graphs

Determine whether the crossing number of the complete graph satisfies \(\operatorname{cr}(K_n)=H(n)\) for every positive integer \(n\), where \(H(n)=\frac14\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor\lfloor(n-2)/2\rfloor\lfloor(n-3)/2\rfloor\).

Background

Hill’s conjecture concerns the minimum number of crossings in a drawing of the complete graph $K_n$. The quantity $H(n)$ comes from explicit drawings constructed by Hill and subsequently published by Guy and by Harary–Hill. The paper notes that the conjecture is known through order fourteen and that an asymptotic lower bound close to $H(n)$ is known for arbitrary $n$, but the universal equality remains the conjectural statement motivating the broader rotation-system problem.

The paper explains that Archdeacon’s conjecture would imply Hill’s conjecture: every simple drawing induces a rotation system, and the number of crossings equals the number of non-planar four-element subsets in that rotation system. Thus, a lower bound of $H(n)$ for all rotation systems would in particular establish the corresponding lower bound for drawings of $K_n$.