The strong chromatic index of $(3,\Delta)$-bipartite graphs

TL;DR

This paper proves that bipartite graphs with one part of maximum degree 3 and the other part of maximum degree Δ can be strongly edge-colored with at most 3Δ colors, confirming a conjecture for this class.

Contribution

It establishes an upper bound of 3Δ colors for strong edge-coloring in (3,Δ)-bipartite graphs, confirming a conjecture for this specific class.

Findings

Strong edge-coloring with at most 3Δ colors for (3,Δ)-bipartite graphs

Confirmation of Brualdi and Quinn Massey's conjecture for this class

Improved understanding of coloring bounds in bipartite graphs

Abstract

A strong edge-coloring of a graph $G=(V,E)$ is a partition of its edge set $E$ into induced matchings. We study bipartite graphs with one part having maximum degree at most $3$ and the other part having maximum degree $\Delta$. We show that every such graph has a strong edge-coloring using at most $3 \Delta$ colors. Our result confirms a conjecture of Brualdi and Quinn Massey ~\cite{[BQ]} for this class of bipartite graphs.