Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs

An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a^{\mathrm{\prime }}(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamčík conjectured that $a^{\mathrm{\prime }}(G)\le \mathrm{\Delta }+2$ for any graph $G$ with maximum degree $\mathrm{\Delta }$. A graph $G$ is said to be $k$-degenerate if every subgraph of $G$ has a vertex of degree at most $k$. Basavaraju and Chandran proved that the conjecture is true for $2$-degenerate graphs. We prove that for a $3$-degenerate graph $G$, $a^{\mathrm{\prime }}(G)\le \mathrm{\Delta }+5$, thereby bringing the upper bound closer to the conjectured bound. We also consider $k$-degenerate graphs with $k\ge 4$ and give an upper bound for the acyclic chromatic index of the same.