Strong edge-coloring of $(3, \Delta)$-bipartite graphs

TL;DR

This paper proves that bipartite graphs with parts of maximum degrees 3 and Δ can be strongly edge-colored with at most 4Δ colors, confirming a conjecture for this class of graphs.

Contribution

It establishes an upper bound of 4Δ colors for strong edge-coloring in bipartite graphs with specified degree constraints, confirming a longstanding conjecture.

Findings

Strong 4Δ-edge-coloring always exists for the specified bipartite graphs.

Confirms a conjecture by Faudree et al. for this class of graphs.

Complements previous results by Steger and Yu.

Abstract

A strong edge-coloring of a graph $G$ is an assignment of colors to edges such that every color class induces a matching. We here focus on bipartite graphs whose one part is of maximum degree at most $3$ and the other part is of maximum degree $\Delta$. For every such graph, we prove that a strong $4\Delta$-edge-coloring can always be obtained. Together with a result of Steger and Yu, this result confirms a conjecture of Faudree, Gy\'arf\'as, Schelp and Tuza for this class of graphs.