# Normal 6-edge-colorings of some bridgeless cubic graphs

## TL;DR

This paper proves that certain classes of bridgeless cubic graphs can be normally edge-colored with six colors, advancing understanding related to Jaeger's Petersen Coloring Conjecture and related graph theory problems.

## Contribution

The authors demonstrate that claw-free bridgeless cubic graphs, permutation snarks, and tree-like snarks admit a normal 6-edge-coloring, improving previous bounds.

## Key findings

- Claw-free bridgeless cubic graphs have a normal 6-edge-coloring.
- Permutation snarks admit a normal 6-edge-coloring.
- At least 7/9 of the edges in any bridgeless cubic graph are normal under some 6-coloring.

## Abstract

In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring if it is rich or poor in this coloring. A normal $k$-edge-coloring of a cubic graph is an edge-coloring with $k$ colors such that each edge of the graph is normal. We denote by $\chi'_{N}(G)$ the smallest $k$, for which $G$ admits a normal $k$-edge-coloring. Normal edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. It is known that proving $\chi'_{N}(G)\leq 5$ for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture. Moreover, Jaeger was able to show that it implies classical conjectures like Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Recently, two of the authors were able to show that any simple cubic graph admits a normal $7$-edge-coloring, and this result is best possible. In the present paper, we show that any claw-free bridgeless cubic graph, permutation snark, tree-like snark admits a normal $6$-edge-coloring. Finally, we show that any bridgeless cubic graph $G$ admits a $6$-edge-coloring such that at least $\frac{7}{9}\cdot |E|$ edges of $G$ are normal.