Acyclic vertex coloring of graphs of maximum degree 5

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of a graph G = (V,E). For a family F of graphs, the acyclic chromatic number of F , denoted by a(F ), is defined as the maximum a(G) over all the graphs G ∈ F . In this paper we show that a(F) = 12, where F is the family of graphs of maximum degree 6 by presenting a linear time algorithm to achieve this bound.