Induced odd cycle packing number, independent sets, and chromatic number

Abstract

AbstractThe induced odd cycle packing number of a graph is the maximum integer such that contains an induced subgraph consisting of 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 Vapnik–Chervonenkis (VC) dimension, and linear independence number admit a randomized efficient polynomial‐time approximation scheme for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers and and any ‐vertex graph of induced odd cycle packing number at most returns in time an independent set of whose size is at least with high probability. In addition, we present ‐boundedness results for graphs with bounded odd cycle packing number, and use them to design a quasipolynomial‐time approximation scheme for the independence number only assuming bounded induced odd cycle packing number.