On the divisibility of graphs Original Research Article

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 V1,…,Vk such that no Vi 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.