Normal edge-colorings of cubic graphs

TL;DR

This paper proves that any simple cubic graph can be normally edge-colored with at most 7 colors, improving previous bounds and linking the problem to flows in 4-edge-connected graphs.

Contribution

It establishes the tight upper bound of 7 for the normal edge-coloring number of all simple cubic graphs, extending prior results and connecting to flow theory.

Findings

Proved that $ ext{chi'}_N(G) \\leq 7$ for all simple cubic graphs.

Connected the existence of specific flows to normal edge-colorings.

Improved the general upper bound from 9 to 7.

Abstract

A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors having the additional property that when looking at the set of colors assigned to any edge $e$ and the four edges adjacent it, we have either exactly five distinct colors or exactly three distinct colors. We denote by $\chi'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal $k$-edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. More precisely, it is known that proving $\chi'_{N}(G)\leq 5$ for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture and then, among others, Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the larger class of all simple cubic graphs (not necessarily bridgeless), some interesting questions naturally arise. For instance, there exist simple cubic graphs, not bridgeless, with $\chi'_{N}(G)=7$. On the other hand, the known best general upper bound for $\chi'_{N}(G)$ was $9$. Here, we improve it by proving that $\chi'_{N}(G)\leq7$ for any simple cubic graph $G$, which is best possible. We obtain this result by proving the existence of specific no-where zero $\mathbb{Z}_2^2$-flows in $4$-edge-connected graphs.