Abstract
A graph G is k- divisible if for each induced subgraph H of G with at least one edge, there is a partition of the vertex set of H into sets V 1,…, V k such that no V i contains a maximum clique of H. We show that a claw-free graph is 2-divisible if and only if it does not contain an odd hole: we conjecture that this result is true for any graph, and present further conjectures relating 2-divisibility to the strong perfect graph conjecture. We also present related results involving the chromatic number and the stability number, with connections to Ramsey theory.
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