Acyclic and Star Colorings of Cographs

TL;DR

This paper proves that for cographs, acyclic and star colorings are equivalent and provides a linear-time algorithm to find optimal colorings, also establishing equalities among several graph parameters.

Contribution

It demonstrates the equivalence of acyclic and star colorings in cographs and offers a linear-time algorithm for optimal coloring based on cotree representation.

Findings

Acyclic and star chromatic numbers are equal for cographs.

A linear-time algorithm exists for optimal acyclic and star coloring of cographs.

Acyclic chromatic number equals the star chromatic number, treewidth plus one, and pathwidth plus one for cographs.

Abstract

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.