Lifted, projected and subgraph-induced inequalities for the representatives k-fold coloring polytope

• Published in: Discrete optimization Vol. 21; pp. 131 - 156
• Main Authors: Campêlo, Manoel, Moura, Phablo F.S., Santos, Marcio C.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2016
• Subjects: ([formula omitted]-fold) graph coloring; Facet; Lexicographic product; Stable set polytope; Web graph; Lexicographic product; Stable set polytope; Web graph; (k-fold) graph coloring; Facet
• ISSN: 1572-5286; 1873-636X
• DOI: 10.1016/j.disopt.2016.06.003

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.