Acyclic Edge Coloring of Triangle Free Planar Graphs

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by $a^{\mathrm{\prime }}(G)$. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that $a^{\mathrm{\prime }}(G)\le \mathrm{\Delta }+2$, where $\Delta =\Delta(G)$ denotes the maximum degree of the graph. If every induced subgraph $H$ of $G$ satisfies the condition $\vert E(H) \vert \le 2\vert V(H) \vert -1$,wesaythatthegraph$G$satisfies$Property
A$\mathrm{.}Inthispaper,weprovethatif$G$satisfies$Property\ A$,then$a’(G)\le \Delta + 3$\mathrm{.}Trianglefreeplanargraphssatisfy$Property\ A$\mathrm{.}Weinferthat$a’(G)\le \Delta + 3$,if$G$ is a triangle free planar graph. Another class of graph which satisfies $PropertyA$ is 2-fold graphs (union of two forests).