Strong edge coloring of subcubic bipartite graphs

A strong edge coloring of a graph $G$ is a proper edge coloring in which each color class is an induced matching of $G$. In 1993, Brualdi and Quinn Massey proposed a conjecture that every bipartite graph without $4$-cycles and with the maximum degrees of the two partite sets $2$ and $\mathrm{\Delta }$ admits a strong edge coloring with at most $\mathrm{\Delta }+2$ colors. We prove that this conjecture holds for such graphs with $\mathrm{\Delta }=3$. We also confirm the conjecture proposed by Faudree et al. for subcubic bipartite graphs.