Testing k-colorability

Let G be a graph on n vertices and suppose that at least εn² edges have to be deleted from it to make it k-colorable. It is shown that in this case most induced subgraphs of G on ck ln k/ε² vertices are not k-colorable, where c > 0 is an absolute constant. If G is as above for k = 2, then most induced subgraphs on (ln(1/ε))^b/ε are nonbipartite, for some absolute positive constant b, and this is tight up to the polylogarithmic factor. Both results are motivated by the study of testing algorithms for k-colorability, first considered by Goldreich, Goldwasser, and Ron in [J. ACM, 45 (1998), pp. 653-750], and improve the results in that paper.