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 χ k ( G ) , is the smallest x such that G admits a k-fold x-coloring. We present an ILP formulation to determine χ k ( G ) and study the facial structure of the corresponding polytope P k ( G ) . We show facets that P k + 1 ( G ) inherits from P k ( G ) . We also relate P k ( G ) to P 1 ( G ∘ K k ) , where G ∘ K k is the lexicographic product of G by a clique with k vertices. In both cases, we can obtain facet-defining inequalities from many of those known for the 1-fold coloring polytope. In addition, we present a class of facet-defining inequalities based on strongly χ k -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.