  An acyclic edge coloring of a graph is a proper edge coloring without any
bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by
$a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$
colors. Fiam\v{c}\'{\i}k conjectured that $a'(G) \le \Delta+2$ for any graph
$G$ with maximum degree $\Delta$. A graph $G$ is said to be $k$-degenerate if
every subgraph of $G$ has a vertex of degree at most $k$. Basavaraju and
Chandran proved that the conjecture is true for $2$-degenerate graphs. We prove
that for a $3$-degenerate graph $G$, $a'(G) \le \Delta+5$, thereby bringing the
upper bound closer to the conjectured bound. We also consider $k$-degenerate
graphs with $k \ge 4$ and give an upper bound for the acyclic chromatic index
of the same.
