  A graph is \textit{locally irregular} if the neighbors of every vertex $v$
have degrees distinct from the degree of $v$. \textit{locally irregular
edge-coloring} of a graph $G$ is an (improper) edge-coloring such that the
graph induced on the edges of any color class is locally irregular. It is
conjectured that $3$ colors suffice for a locally irregular edge-coloring.
Recently, Bensmail et al. (Bensmail, Merker, Thomassen: Decomposing graphs into
a constant number of locally irregular subgraphs, {\em European J. Combin.},
60:124--134, 2017) settled the first constant upper bound for the problem to
$328$ colors. In this paper, using a combination of existing results, we
present an improvement of the bounds for bipartite graphs and general graphs,
setting the best upper bounds to $7$ and $220$, respectively. In addition, we
also prove that $4$ colors suffice for locally irregular edge-coloring of any
subcubic graph.
