Induced odd cycle packing number, independent sets, and chromatic number
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The induced odd cycle packing numberiocp(G) of a graph G is the maximum integer k such that G contains an induced subgraph consisting of k pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al. proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers k> 0 and t> 1 and any n-vertex graph G of induced odd cycle packing number returns in time O_k,t(n^k+4) an independent set of G whose size is at least α(G)-n/t with high probability. In addition, we present χ-boundedness results for graphs with bounded odd cycle packing number, and use them to design a QPTAS for the independence number only assuming bounded induced odd cycle packing number.
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