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 give complete characterizations of the associated polytopes for trees and block graphs. Also, we show that the vertex coloring polytope associated to this formulation corresponds to a face of the stable set polytope of a particular graph and, based on this fact, we derive a new family of valid inequalities generalizing several known families from the literature. We conjecture that this new family of valid inequalities is sufficient to completely describe the vertex coloring polytope associated to cycles.