  An acyclic edge coloring of a graph is a proper edge coloring in which there
are no bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted
by $a'(G)$, is the minimum positive integer $k$ such that $G$ has an acyclic
edge coloring with $k$ colors. It has been conjectured by Fiam\v{c}\'{\i}k that
$a'(G) \le \Delta+2$ for any graph $G$ with maximum degree $\Delta$. Linear
arboricity of a graph $G$, denoted by $la(G)$, is the minimum number of linear
forests into which the edges of $G$ can be partitioned. A graph is said to be
chordless if no cycle in the graph contains a chord. Every $2$-connected
chordless graph is a minimally $2$-connected graph. It was shown by Basavaraju
and Chandran that if $G$ is $2$-degenerate, then $a'(G) \le \Delta+1$. Since
chordless graphs are also $2$-degenerate, we have $a'(G) \le \Delta+1$ for any
chordless graph $G$. Machado, de Figueiredo and Trotignon proved that the
chromatic index of a chordless graph is $\Delta$ when $\Delta \ge 3$. They also
obtained a polynomial time algorithm to color a chordless graph optimally. We
improve this result by proving that the acyclic chromatic index of a chordless
graph is $\Delta$, except when $\Delta=2$ and the graph has a cycle, in which
case it is $\Delta+1$. We also provide the sketch of a polynomial time
algorithm for an optimal acyclic edge coloring of a chordless graph. As a
byproduct, we also prove that $la(G) = \lceil \frac{\Delta }{2} \rceil$, unless
$G$ has a cycle with $\Delta=2$, in which case $la(G) = \lceil
\frac{\Delta+1}{2} \rceil = 2$. To obtain the result on acyclic chromatic
index, we prove a structural result on chordless graphs which is a refinement
of the structure given by Machado, de Figueiredo and Trotignon for this class
of graphs. This might be of independent interest.
