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

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.