The Strong Chromatic Index of graphs with maximum degree $Δ$

A strong edge-coloring of a graph $G$ is an edge-coloring such that no two edges of distance at most two receive the same color. The strong chromatic index $χ'_s(G)$ is the minimum number of colors in a strong edge-coloring of $G$. P. Erdős and J. Nešetřil conjectured in 1985 that $χ'_s(G)$ is bounded above by $\frac54Δ^2$ when $Δ$ is even and $\frac14(5Δ^2-2Δ+1)$ when $Δ$ is odd, where $Δ$ is the maximum degree of $G$. In this paper, we give an algorithm that uses at most $2Δ^2-3Δ+2$ colors for graphs with girth at least $5$. And in particular, we prove that any graph with maximum degree $Δ=5$ has a strong edge-coloring with $37$ colors.