Normal 6-edge-colorings of cubic graphs with oddness 2

TL;DR

This paper proves that all cubic graphs with oddness 2 can be normally edge-colored with at most 6 colors, extending previous results and supporting the conjecture that 5 colors suffice for all bridgeless cubic graphs.

Contribution

It extends the class of cubic graphs known to admit a normal 6-edge-coloring to include those with oddness 2.

Findings

Cubic graphs with oddness 2 admit a normal edge-coloring with at most 6 colors.

The result generalizes previous work on cycle permutation graphs.

Supports the conjecture that 5 colors suffice for all bridgeless cubic graphs.

Abstract

A normal edge-coloring of a cubic graph is a proper edge-coloring, in which every edge is adjacent to edges colored with four distinct colors or to edges colored with two distinct colors. It is conjectured that $5$ colors suffice for a normal edge-coloring of any bridgeless cubic graph and this statement is equivalent to the Petersen Coloring Conjecture. In this paper, we extend the result of Mazzuoccolo and Mkrtchyan (Normal 6-edge-colorings of some bridgeless cubic graphs, Discrete Appl. Math. 277 (2020), 252--262), who proved that every cycle permutation graph admits a normal edge-coloring with at most $6$ colors. In particular, we show that every cubic graph with oddness $2$ admits a normal edge-coloring with at most $6$ colors.