Strong edge colorings of graphs and the covers of Kneser graphs

TL;DR

This paper characterizes when a k-regular graph can be strongly edge-colored with 2k-1 colors, linking it to graph covers of Kneser graphs, and disproves a related conjecture for subcubic graphs.

Contribution

It establishes a precise condition connecting strong edge colorings of k-regular graphs to covers of Kneser graphs, providing a new characterization and refuting a previous conjecture.

Findings

A k-regular graph admits a strong edge coloring with 2k-1 colors iff it covers K(2k-1,k-1).

Cubic graphs cover the Petersen graph and are strongly 5-edge-colorable.

A conjecture on strong edge colorings of subcubic graphs is shown to be false.

Abstract

A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph admits a strong edge coloring with $2k-1$ colors if and only if it covers the Kneser graph $K(2k-1,k-1)$. In particular, a cubic graph is strongly $5$-edge-colorable whenever it covers the Petersen graph. One of the implications of this result is that a conjecture about strong edge colorings of subcubic graphs proposed by Faudree et al. [Ars Combin. 29 B (1990), 205--211] is false.