A. Lyons, "Acyclic and Star Colorings of Joins of Graphs and an Algorithm for Cographs," Proc. 8th Cologne_Twente Workshop on Graphs and Combinatorial Optimization, 2009, pp. 199-202. Also Preprint ANL/MCS-P1610-0409, April 2009. [pdf]
An acyclic coloring of a graph is a proper vertex coloring such that the subgraph induced by the union of any two color classes is a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. The acyclic and star chromatic numbers of a graph G are defined analogously to the chromatic number !(G) and are denoted by !a(G) and !s(G), respectively. In this paper, we consider acyclic and star colorings of graphs that are decomposable with respect to the join operation, which builds a new graph from a collection of two or more disjoint graphs by adding all possible edges between them. In particular, we present a recursive formula for the acyclic chromatic number of joins of graphs and show that a similar formula holds for the star chromatic number. We also demonstrate the algorithmic implications of our results for the cographs, which have the unique property that they are recursively decomposable with respect to the join and disjoint union operations.