Polyhedral studies of vertex coloring problems: The standard formulation

Despite the fact that many vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not “under control” from a polyhedral point of view. The equivalence between optimization and separation suggests the existence of integer programming formulations for these problems whose associated polytopes admit elegant characterizations. In this work we address this issue. As a starting point, we focus our attention on the well-known standard formulation   for the classical vertex coloring problem. We present some general results about this formulation and we show that the vertex coloring polytope associated to this formulation for a graph GG and a set of colors CC corresponds to a face of the stable set polytope   of a particular graph SGC. We further study the perfectness of SGC showing that when ∣C∣>2∣C∣>2, this graph is perfect if and only if GG is a block graph, from which we deduce a complete characterization of the associated coloring polytopes for block graphs. We also derive a new family of valid inequalities generalizing several known families from the literature and we conjecture that this family is sufficient to completely describe the vertex coloring polytope associated to cacti graphs.