Discoveries / combinatorics

Petersen Coloring Conjecture

This preprint was not the first disproof. A 68-vertex counterexample was posted to X on 23 July 2026 by @NeuralReformist, credited to GPT-5.6 Sol Ultra, sixteen days earlier. This site decoded that sparse6 string and checked it independently: 68 vertices, 102 edges, simple, cubic, connected, bridgeless, girth five, and no Petersen coloring under the same encoder used for the 112-vertex graph. Whether the two are independent is unknown - the preprint does not cite the post. The headline axes still record the preprint, the only complete writeup with certificates. The implication runs one way: the Petersen coloring conjecture implies Berge-Fulkerson and the 5-cycle-double-cover conjecture, so refuting it leaves both of those open. The paper does not claim 112 is minimum, and it supplies a second, nonisomorphic $D_3$-symmetric 112-vertex counterexample. Combined with a theorem of Ma, Mattiolo, Steffen and Wolf, one counterexample yields infinitely many.