  A strong edge-coloring of a graph $G$ is a coloring of the edges such that
every color class induces a matching in $G$. The strong chromatic index of a
graph is the minimum number of colors needed in a strong edge-coloring of the
graph. In 1985, Erd\H{o}s and Ne\v{s}et\v{r}il conjectured that every graph
with maximum degree $\Delta$ has a strong edge-coloring using at most
$\frac{5}{4}\Delta^2$ colors if $\Delta$ is even, and at most
$\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd.
Despite recent progress for large $\Delta$ by using an iterative probabilistic
argument, the only nontrivial case of the conjecture that has been verified is
when $\Delta = 3$, leaving the need for new approaches to verify the conjecture
for any $\Delta\ge 4$. In this paper, we apply some ideas used in previous
results to an upper bound of 21 for graphs with maximum degree 4, which
improves a previous bound due to Cranston in 2006 and moves closer to the
conjectured upper bound of 20.
