On the representatives k-fold coloring polytope

A k-fold x-coloring of a graph G is an assignment of (at least) k distinct colors from the set {1, 2, . . . , x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The k-th chromatic number of G, denoted by χk(G), is the smallest x such that G admits a k-fold x-coloring. We present an ILP formulation to determine χk(G) and study the facial structure of the corresponding polytope Pk(G). We show facets that Pk+1(G) inherits from Pk(G). We also relate Pk(G) to P1(G◦Kk), whereG◦Kk is the lexicographic product ofG by a clique with k vertices. In both cases, we can obtain facet-defining inequalities from most of those known for the 1-fold coloring polytope. In addition, we present a class of facet-defining inequalities based on strongly χk-critical webs, which extend and generalize known corresponding results for 1-fold coloring. We introduce this criticality concept and characerize the webs having such a property.