Total Coloring and Total Matching: Polyhedra and Facets

A total coloring of a graph G=(V,E) is an assignment colors to vertices and edges such that neither two adjacent nor incident get the same color, and, for each edge, end-points edge itself receive different colors. Any valid induces partition elements G into matchings, which are defined as subsets can take color. In this paper, we propose Integer Linear Programming models both Total Coloring Matching problems, study strength corresponding relaxations. The formulated problem finding minimum number matchings cover all elements. This covering formulation be solved by Column Generation algorithm, where pricing subproblem corresponds Weighted Problem. Hence, Polytope. We introduce three families nontrivial inequalities: vertex-clique inequalities based on standard clique Stable Set Polytope, congruent-2k3 cycle parity vertex set induced cycle, even-clique complete subgraphs even order. prove facet-defining only when k=4, while even-cliques always facet-defining. Finally, present preliminary computational results algorithm Problem Cutting Plane