Acyclic Edge coloring of 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 $\Delta =\Delta(G)$denotesthemaximumdegreeofthegraph\mathrm{.}Weprovethatif$G$ is a planar graph with maximum degree $\mathrm{\Delta }$, then $a^{\mathrm{\prime }}(G)\le \mathrm{\Delta }+12$.