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 $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiam\v{c}\'{\i}k conjectured that $a'(G) \le \Delta+2$ for any graph $G$ with maximum degree $\Delta$. 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 $G$, $a'(G) \le \Delta+5$, 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.

翻译：图的 无环边染色 是指一种不含双色环路的正常边染色。图 \(G\) 的 无环色指数，记作 \(a'(G)\)，是使得 \(G\) 存在用 \(k\) 种颜色的无环边染色的最小 \(k\) 值。Fiam\v{c}\'{\i}k 猜想：对于任意最大度为 \(\Delta\) 的图 \(G\)，有 \(a'(G) \le \Delta+2\)。若图 \(G\) 的每个子图均存在度数不超过 \(k\) 的顶点，则称 \(G\) 为 \(k\)-退化图。Basavaraju 和 Chandran 证明了该猜想对 2-退化图成立。本文证明：对于 3-退化图 \(G\)，有 \(a'(G) \le \Delta+5\)，从而将上界进一步逼近猜想值。此外，我们考虑 \(k \ge 4\) 的 \(k\)-退化图，并给出其无环色指数的一个上界。