Nonexistence of Whirling-Knight Tours at Half Coil Count for $n \equiv 4, 6 \pmod 8$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915999778865152 |
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| author | Li, Shisheng |
| author_facet | Li, Shisheng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nonexistence of Whirling-Knight Tours at Half Coil Count for $n \equiv 4, 6 \pmod 8$ Li, Shisheng Combinatorics 05C45, 05C38, 05C20, 90C05 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. |
| title | Nonexistence of Whirling-Knight Tours at Half Coil Count for $n \equiv 4, 6 \pmod 8$ |
| topic | Combinatorics 05C45, 05C38, 05C20, 90C05 |
| url | https://arxiv.org/abs/2605.04603 |