math.CO — Combinatorics

Two conjectures of Wormald on Hamilton cycles in tournaments

Contributed by Deep Bhattacharjee

For a regular tournament on $d$ vertices with skew adjacency matrix $S$, the Hessian determinant $H$ that Wormald attached to its random blow-ups equals $d\det(d^2I+S^2)/\det(d^2I-S^2)$. Hence $H\neq0$ and $d/H$ is the square of a rational number, as Wormald conjectured. For the cyclic tournament it gives $\sqrt{d/H}=\bigl((d+1)^d+(d-1)^d\bigr)/\bigl((d+i)^d+(d-i)^d\bigr)$, so the associated constants tend to $\cosh 1/\cos 1=2.8559578925\ldots$; this proves a second conjecture of Wormald and identifies his numerical constant.