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A coloring of a graph is an assignment of colors to its vertices such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring is a coloring such that no cycle 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 a previous work we presented facet-inducing families of valid inequalities based on induced even cycles for the polytope associated to an integer programming formulation of the acyclic coloring problem. In this work we continue with this study by introducing new families of facet-inducing inequalities based on combinations of even cycles and cliques.

In a previous work we presented six facet-inducing families of valid inequalities for the polytope associated to an integer programming formulation of the acyclic coloring problem. In this work we study their disjunctive rank, as defined by [E. Balas, S. Ceria and G. Cornuejols, Math. Program. 58 (1993) 295–324]. We also propose to study a dual concept, which we call the disjunctive anti-rank of a valid inequality.