Nonexistence of Whirling-Knight Tours at Half Coil Count for $n \equiv 4, 6 \pmod 8$

TL;DR

The paper proves the nonexistence of certain Hamiltonian cycles called whirling knight tours with specific coil counts on n x n boards for particular values of n, settling a conjecture by Beluhov.

Contribution

It establishes the nonexistence of whirling knight tours with coil count n/2 for n ≡ 4 or 6 mod 8, using novel infeasibility certificates for each residue class.

Findings

No whirling knight tours with c = n/2 exist for n ≡ 4, 6 mod 8.

Constructed explicit infeasibility certificates for each residue class.

Settled a conjecture of Beluhov regarding these tours.

Abstract

A whirling knight's tour is a Hamiltonian cycle in the digraph of counter-clockwise knight steps about the centre of an $n \times n$ board; its coil count $c$ is the winding number around the centre. We prove that no such tour with $c = n/2$ exists when $n \equiv 4 \pmod 8$ ($n \ge 4$) or $n \equiv 6 \pmod 8$ ($n \ge 6$), settling a conjecture of Beluhov. For each residue class we exhibit a closed-form Farkas certificate for infeasibility of a cycle-cover LP relaxation; the two certificates are structurally distinct.