Acyclic coloring of graphs with maximum degree at most six

Abstract

An acyclic k-coloring of a graph G is a proper vertex coloring using k colors such that any cycle of G is colored with at least three colors, and the minimum integer k is called the acyclic chromatic number of G, denoted by χa(G). Grünbaum conjectured that the acyclic chromatic number of graphs with maximum degree at most Δ is no more than Δ+1 in 1973. By now, the conjecture has been proved only for Δ≤4. We show that χa(G)≤9 for Δ≤6, which improves the result χa(G)≤10 of Zhao et al. (2014).