Edge-3-Coloring of a Family of Cubic Graphs

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.