Disproving the Petersen Coloring Conjecture: Theoretical Analysis and an Infinite Family of Counterexamples

Abstract

In 1988, Jaeger conjectured that every bridgeless cubic graph $G$ admits a Petersen coloring; that is, a map $E(G) \to E(P)$ mapping any two adjacent edges of $G$ to two adjacent edges of the Petersen graph $P$. A positive resolution to Jaeger's conjecture would have immediately resolved several other famous and long-standing problems in graph theory. In July 2026, a 68-vertex counterexample was announced on X. Shortly afterwards, Putman independently presented two non-isomorphic 112-vertex counterexamples, relying solely on computer-assisted verification. In this paper, we present two counterexamples of order $52$, currently the smallest known, and provide a purely theoretical proof. In the second part, we construct an infinite family of cyclically $4$-edge-connected cubic graphs without a Petersen coloring for every even order at least $60$. Additionally, through computational verification, we show that any counterexample must have order at least $40$. Moreover, we also show that our counterexamples provide a negative answer to other related problems. Finally, we conclude the paper by discussing key open problems and highlighting avenues for future work.