Partially normal 5-edge-colorings of cubic graphs

TL;DR

This paper advances understanding of 5-edge-colorings in cubic graphs by showing most edges can be normal, supporting the Petersen coloring conjecture, and improving previous bounds on normal edges.

Contribution

It proves that in bridgeless cubic graphs, at least 80% of edges can be normal in a proper 5-edge-coloring, refining earlier bounds and related to the Petersen coloring conjecture.

Findings

At least 80% of edges are normal in the coloring.

Improves previous bounds on the number of normal edges.

Supports the Petersen coloring conjecture with a new partial result.

Abstract

In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal 5-edge-coloring, that is, a proper 5-edge-coloring such that all edges are normal. In this paper, we prove a result related to the Petersen coloring conjecture. The parameter $\mu_3$ is a measurement for cubic graphs, introduced by Steffen in 2015. Our result shows that every bridgeless cubic graph $G$ has a proper 5-edge-coloring such that at least $|E(G)|-\mu_3(G)$, which is no less than $\frac{4}{5}|E(G)|$, many edges are normal. This result improves on some earlier results of B\'{\i}lkov\'{a} and \v{S}\'{a}mal.