Disjunctive ranks and anti-ranks of some facet-inducing inequalities of the acyclic coloring polytope M´onicaBraga 1,2 Javier Marenco 1,3 Sciences Institute, National University of General Sarmiento Malvinas Argentinas, Argentina Abstract 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 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. Keywords: polyhedral combinatorics, acyclic coloring, disjunctive rank 1 Partially supported by ANPCyT Grant PICT 2007-00518 and UBACyT Grant X069. 2 Email: mbraga@ungs.edu.ar 3 Email: jmarenco@ungs.edu.ar Electronic Notes in Discrete Mathematics 37 (2011) 213–218 1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm doi:10.1016/j.endm.2011.05.037