Edge-$3$-Coloring of a Family of Cubic Graphs

Abstract

Let $G$ be a cubic graph containing no subdivision of the Petersen graph. If $G$ has a $2$-factor $F$ consisting of two circuits $C_1$ and $C_2$ such that $C_1$ is chordless and $C_2$ has at most one chord, then $G$ is edge-$3$-colorable.This result generalizes an early result by Ellingham and is a partial result of Tutte’s edge-$3$-coloring conjecture.