A Strong Edge-Coloring of Graphs with Maximum Degree 4 Using 22 Colors

In 1985, Erdős and Neśetril conjectured that the strong edge-coloring number of a graph is bounded above by ${5/4}Δ^2$ when $Δ$ is even and ${1/4}(5Δ^2-2Δ+1)$ when $Δ$ is odd. They gave a simple construction which requires this many colors. The conjecture has been verified for $Δ\leq 3$. For $Δ=4$, the conjectured bound is 20. Previously, the best known upper bound was 23 due to Horak. In this paper we give an algorithm that uses at most 22 colors.

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