HEXAGONbeta

Two conjectures of Wormald on Hamilton cycles in tournaments

A mix of human-written and AI-generated textHuman understanding: some partsmath.CO — Combinatoricsmath.RA — Rings and Algebras

Contributed by Deep Bhattacharjee ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

For a regular tournament on dd vertices with skew adjacency matrix SS, the Hessian determinant HH that Wormald attached to its random blow-ups equals ddet⁡(d2I+S2)/det⁡(d2I−S2)d\det(d^2I+S^2)/\det(d^2I-S^2). Hence H≠0H\neq0 and d/Hd/H is the square of a rational number, as Wormald conjectured. For the cyclic tournament it gives d/H=((d+1)d+(d−1)d)/((d+i)d+(d−i)d)\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…\cosh 1/\cos 1=2.8559578925\ldots; this proves a second conjecture of Wormald and identifies his numerical constant.

Provenance statement

Deep Bhattacharjee chose the problem and directed the work. Claude (Anthropic), through Claude Code, carried out the searches and computations, wrote the checking programs and the Lean 4 proofs and drafted the paper. Settles Wormald's Conjectures 2 and 6. The proofs are written out in the paper; the programs and their outputs are in the release. The author approved the final version. Code and data: release v1.0.0, https://github.com/creelie/Breakthrough-problems-1/releases/tag/v1.0.0.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 1 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositCurrentOct 22, 2026