Acyclic coloring of claw-free graphs with small degree

An acyclic -coloring of a graph is a proper vertex coloring using colors such that every cycle is colored with at least three colors. A graph is claw-free if it contains no as an induced subgraph. Grünbaum (1973) conjectured that every graph with maximum degree has an acyclic -coloring. In this paper, we show that this conjecture holds for the claw-free graphs with .