Acyclic coloring of claw-free graphs with small degree

An acyclic k-coloring of a graph is a proper vertex coloring using k colors such that every cycle is colored with at least three colors. A graph is claw-free if it contains no K-1,K-3 as an induced subgraph. Grunbaum (1973) conjectured that every graph with maximum degree Delta has an acyclic (Delta + 1)-coloring. In this paper, we show that this conjecture holds for the claw-free graphs with Delta <= 6. (c) 2022 Elsevier B.V. All rights reserved.