  An $acyclic$ edge coloring of a graph is a proper edge coloring such that
there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph
is the minimum number k such that there is an acyclic e dge coloring using k
colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks
that for any simple and finite graph $G$, $a'(G)\le \Delta+2$, where $\Delta
=\Delta(G)$ denotes the maximum degree of $G$. We prove the conjecture for
connected graphs with $\Delta(G) \le 4$, with the additional restriction that
$m \le 2n-1$, where $n$ is the number of vertices and $m$ is the number of
edges in $G $. Note that for any graph $G$, $m \le 2n$, when $\Delta(G) \le 4$.
It follows that for any graph $G$ if $\Delta(G) \le 4$, then $a'(G) \le 7$.
