Disjunctive ranks and anti-ranks of some facet-inducing inequalities of the acyclic coloring polytope

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 χA(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 χA(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 study the disjunctive rank of six facet-inducing families of valid inequalities for the polytope associated to a natural integer programming formulation of the acyclic coloring problem. We also introduce the concept of disjunctive anti-rank and study the anti-rank of these families.