Acyclic and Star Colorings of Cographs

An \emph{acyclic coloring} of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of \emph{star coloring} requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding an optimal acyclic and star coloring of a cograph. If the graph is given in the form of a cotree, the algorithm runs in O(n) time. We also show that the acyclic chromatic number, the star chromatic number, the treewidth plus one, and the pathwidth plus one are all equal for cographs.