Strong chromatic index of bipartite graphs

An edge-coloring of a graph $G$ is called a strong edge-coloring if all its color classes are induced matchings in $G$; the minimum number of colors required for such a coloring, denoted by ${\chi }_s^{\mathrm{\prime }}(G)$, is known as the strong chromatic index of $G$. For each vertex $v$ of a graph $G$, let $d_G(v)$ denote the degree of $v$ in $G$. Let $G$ be a bipartite graph with partite sets $A$ and $B$, and let ${\mathrm{\Delta }}_A=\max \{d_G(a):a\in A\}$ and ${\mathrm{\Delta }}_B=\max \{d_G(b):b\in B\}$. A conjecture of Brualdi and Quinn Massey asserts that ${\chi }_s^{\mathrm{\prime }}(G)\le {\mathrm{\Delta }}_A{\mathrm{\Delta }}_B$. In this paper, we show that ${\chi }_s^{\mathrm{\prime }}(G)\le 1.676{\mathrm{\Delta }}_A{\mathrm{\Delta }}_B$ provided that the product ${\mathrm{\Delta }}_A{\mathrm{\Delta }}_B$ is sufficiently large.