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 kth chromatic number of G, denoted by χk(G), is the smallest x such that G admits a k-fold x-coloring. We present an integer linear programming formulation (ILP) 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) and show how to lift facets from Pk(G) to Pk+ℓ(G). We project facets of P1(G∘Kk) into facets of Pk(G), where G∘Kk 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. We also derive facets of Pk(G) from facets of stable set polytopes of subgraphs of G. In addition, we present classes of facet-defining inequalities based on strongly χk-critical webs and antiwebs, which extend and generalize known results for 1-fold coloring. We introduce this criticality concept and characterize the webs and antiwebs having such a property.