# Hardness of finding independent sets in almost q-colorable graphs

## Abstract

We show that for any ε > 0, and positive integers k and q such that q ≥= 2k + 1, given a graph on N vertices that has a q-colorable induced sub graph of (1 - ε)N vertices, it is NP-hard to find an independent set of N/qk+1 vertices. This substantially improves upon the work of Dinur et al. [DKPS] who gave a corresponding bound of N/q2. Our result implies that for any positive integer k, given a graph that has an independent set of ≈ (2k + 1)-1 fraction of vertices, it is NP-hard to find an independent set of (2k + 1)-(k+1) fraction of vertices. This improves on the previous work of Engebretsen and Holmerin [EH] who proved a gap of ≈ 2-k vs 2-(k 2), which is best possible using techniques (including those of [EH]) based on the query efficient PCP of Samorodnitsky and Trevisan [3]. © 2012 IEEE.