Machine-checked certificates for Haugland's 2131-vertex Moser-spindle-free 5-chromatic unit-distance graph, and a certified reduction to 1501 vertices

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-21
DOI
https://doi.org/10.5281/zenodo.22878890
Primary Topic
Advanced Graph Theory Research
Type
preprint
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preprint

Machine-checked certificates for Haugland's 2131-vertex Moser-spindle-free 5-chromatic unit-distance graph, and a certified reduction to 1501 vertices

King Tat Wong
Zenodo (CERN European Organization for Nuclear Research)
Advanced Graph Theory Research
preprint

Machine-checked certificates for Haugland's 2131-vertex Moser-spindle-free 5-chromatic unit-distance graph, and a certified reduction to 1501 vertices

King Tat Wong
preprint en

Abstract

Haugland (arXiv:2608.04542) constructed a unit-distance graph G₃ in the plane with 2131 vertices that contains no Moser spindle as a subgraph and has chromatic number 5. The 5-chromaticity rests on a SAT computation for an auxiliary 740-vertex graph G₁ (the "pair property"), for which no proof artifact was published. We reconstruct the construction from the paper's definitions in exact cyclotomic arithmetic (all counts match), and give machine-checkable certificates for every step: the edge lists are the complete unit-distance graphs on the exact point sets; G₃ contains no Moser spindle (a DRAT-certified refutation of a subgraph-containment encoding); G₃ is 5-colourable; and G₃ is not 4-colourable, via a three-part chain L1–L3 in which L1, the pair property of G₁, is certified by cube-and-conquer: march_cu splits the instance into 14 786 cubes, each refuted by kissat with a drat-trim-verified proof (5 % also checked by the formally verified cake_lpr), together with a verified cover certificate. Twenty-two plain CDCL configurations did not settle L1 within 30 minutes each, whereas the cube-and-conquer campaign took 16 minutes on 14 cores. We also report reduction experiments. Deleting certified sets of 30 and then 60 vertices from G₁ preserves the pair property and yields Moser-spindle-free 5-chromatic graphs on 2011 and 1891 vertices respectively, while the next batch of 30 cannot be removed as a whole. A longer campaign on the auxiliary graph G₂, run in certified batches whose size was reduced from 30 to 20 to 10 to 5, reaches a Moser-spindle-free 5-chromatic unit-distance graph on 1501 vertices and 8088 edges, a reduction of 630 vertices (29.6 %) from Haugland's graph; the campaign stops for a structural reason rather than a budget one, with both independent candidate batches at step size 5 refuted by explicit 4-colourings. This is still larger than Heule's 1441-vertex example, but every step of the chain is certified and independently re-checkable. The inputs, cubes, top-level certificates and a re-verification script are released (GitHub and Zenodo, DOI 10.5281/zenodo.22435777); the 128 GB of leaf proofs for the 1501-vertex graph can be regenerated and re-checked from these rather than being deposited.

Zenodo (CERN European Organization for Nuclear Research)
Advanced Graph Theory Research
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