Acyclic Edge Coloring of 3-sparse Graphs
Advances the understanding of acyclic edge coloring for a specific graph class, but is incremental as it addresses a restricted subclass of graphs.
The paper proves Fiamčík's conjecture that acyclic edge chromatic number ≤ Δ+2 for 3-sparse graphs, and gives a stronger bound of Δ+1 under an additional condition.
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 $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic edge coloring with $k$ colors. FiamÄ\'ık conjectured that for a graph $G$ with maximum degree $Î$, $a'(G) \le Î+2$. 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 $Î+1$, if there exists an edge $xy$ in the graph with $d_G(x)+ d_G(y) < Î+3$. When $ Î> 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 $Î$.