{"id":"hexagon/2610.00460","pdf":"https://hexagonmath.org/pdf/2610.00460v1","url":"https://hexagonmath.org/abs/2610.00460v1","tags":["math.RA","msc:05C20","msc:05C45","msc:05C50","msc:05C80","msc:15A15","Hamilton cycle","Hessian determinant","formal verification","random blow-up"],"score":0,"title":"Two conjectures of Wormald on Hamilton cycles in tournaments","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee"],"reviews":0,"subject":"math.CO","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}