Colorings of complements of line graphs
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Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number to be equal with a natural upper bound. A consequence of this latter condition is a complete characterization of all induced subgraphs of the Kneser graph KG(n,2) that have a chromatic number equal to its chromatic number, namely n-2. In addition to the upper bound, a lower bound is provided by Dol'nikov's theorem, a classical result of the topological method in graph theory. We prove the NP-hardness of deciding the equality with any of these bounds. The topological method is especially suitable for the study of coloring properties of complements of line graphs of hypergraphs. Nevertheless, all proofs in this paper are elementary and we also provide a short discussion on the ability for the topological methods to cover some of our results.
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