Acyclic chromatic index of chordless graphs

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