Strong Edge-Coloring of Cubic Bipartite Graphs: A Counterexample

TL;DR

This paper presents a new counterexample to a longstanding conjecture that all large girth bipartite cubic graphs can be strongly edge-colored with five colors, thereby disproving the conjecture.

Contribution

The authors provide an alternative construction that disproves the 1990 conjecture about strong edge-coloring of bipartite cubic graphs with large girth.

Findings

Counterexample to the conjecture

Disproof of the 1990 conjecture

Alternative construction method

Abstract

A strong edge-coloring $\varphi$ of a graph $G$ assigns colors to edges of $G$ such that $\varphi(e_1)\ne \varphi(e_2)$ whenever $e_1$ and $e_2$ are at distance no more than 1. It is equivalent to a proper vertex coloring of the square of the line graph of $G$. In 1990 Faudree, Schelp, Gy\'arf\'as, and Tuza conjectured that if $G$ is a bipartite graph with maximum degree 3 and sufficiently large girth, then $G$ has a strong edge-coloring with at most 5 colors. In 2021 this conjecture was disproved by Lu\v{z}ar, Ma\v{c}ajov\'{a}, \v{S}koviera, and Sot\'{a}k. Here we give an alternative construction to disprove the conjecture.