Acyclic Edge Coloring of 3-sparse Graphs

A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a^{\mathrm{\prime }}(G)$, is the minimum integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. Fiamčík conjectured that for a graph $G$ with maximum degree $\mathrm{\Delta }$, $a^{\mathrm{\prime }}(G)\le \mathrm{\Delta }+2$. A graph $G$ is said to be $3$-sparse if every edge in $G$ is incident on at least one vertex of degree at most $3$. We prove the conjecture for the class of $3$-sparse graphs. Further, we give a stronger bound of $\mathrm{\Delta }+1$, if there exists an edge $xy$ in the graph with $d_G(x)+d_G(y)<\mathrm{\Delta }+3$. When $\mathrm{\Delta }>3$, the $3$-sparse graphs where no such edge exists is the set of bipartite graphs where one partition has vertices with degree exactly $3$ and the other partition has vertices with degree exactly $\mathrm{\Delta }$.