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arXiv · 2611.06269

Small 5-chromatic unit-distance graphs in disks of radius 3/2, 5/4 and 23/20

Chandan Shukla
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Abstract

We give three explicit unit-distance graphs in the plane with chromatic number 5 whose vertices lie in small disks: a graph with 773 vertices inside an open disk of radius 3/2, one with 5713 vertices inside an open disk of radius 5/4, and one with 8168 vertices inside an open disk of radius 23/20. The smallest examples previously reported for these radii have 11725, 11725 and 31171 vertices. All coordinates are exact elements of Q(sqrt3, sqrt5, sqrt11). In each case 5-chromaticity is certified by an explicit 5-colouring and a DRAT proof that no 4-colouring exists. The 773-vertex graph is vertex-critical.

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Chandan Shukla. 2026-11-04. Small 5-chromatic unit-distance graphs in disks of radius 3/2, 5/4 and 23/20. https://arxiv.org/abs/2611.06269

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