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Coloring Graphs with Dense Neighborhoods

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Abstract

It is shown that any graph with maximum degree Δ in which the average degree of the induced subgraph on the set of all neighbors of each vertex exceeds math formula is either math formula-colorable or contains a clique on more than math formula vertices. In the math formula case we improve the bound on the average degree to math formula and the bound on the clique number to math formula. As corollaries, we show that every graph satisfies math formula and every graph satisfies math formula.

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