A polyhedral study of the acyclic coloring problem

A coloring of a graph G is an assignment of colors to the vertices of G such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring of G is a coloring such that no cycle of G receives exactly two colors, and the acyclic chromatic number χ A ( G ) of a graph G is the minimum number of colors in any such coloring of G. Given a graph G and an integer k, determining whether χ A ( G ) ⩽ k or not is NP-complete even for k = 3 . The acyclic coloring problem arises in the context of efficient computations of sparse and symmetric Hessian matrices via substitution methods. In this work we start an integer programming approach for this problem, by introducing a natural integer programming formulation and presenting six facet-inducing families of valid inequalities.