A new characterization for parity graphs and a coloring problem with costs
In this paper, we give a characterization for parity graphs. A graph is a parity graph, if and only if for every pair of vertices all minimal chains joining them have the same parity. We prove that G is a parity graph, if and only if the Cartesian product G×K 2 is a perfect graph.
Furthermore, as a consequence we get a result for the polyhedron corresponding to an integer linear program formulation of a coloring problem with costs. For the case that the costs k v,3 =k v,c for each color c ≥ 3 and vertex v ∃ V, we show that the polyhedron contains only integral 0/1 extrema if and only if the graph G is a parity graph.
KeywordsBipartite Graph Maximum Clique Coloring Problem Perfect Graph Integer Linear Program Formulation
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