Smaller 5-Chromatic Unit-Distance Graphs on Two Icosahedral Spheres

Voronov, Neopryatnaya and Dergachev constructed 5-chromatic unit-distance graphs on the sphere \( S^2(r_1), r_1=\cos\frac{\pi}{10} \), with \( 372 \) vertices, and on \( S^2(r_2) \), \( r_2=\cos\frac{3\pi}{10} \), with \( 972 \) vertices, and noted that the first graph stays 5-chromatic after deleting any single vertex. By SAT-based vertex minimization we find a 5-chromatic subgraph of the first graph with \( 231 \) vertices and \( 938 \) edges, and one of the second with \( 961 \) vertices and \( 4028 \) edges. Both are vertex-critical and contain no Moser spindle; the smaller one is the best found by our search, not a proved minimum. All coordinates are given exactly in nested radicals and verified in exact arithmetic; non-4-colourability is certified by DRAT proofs checked with DRAT-TRIM.