Testing κ-colorability

Let G be a graph on n vertices and suppose that at least epsilonn(2) edges have to be deleted fromi t to make it k-colorable. It is shown that in this case most induced subgraphs of G on ck ln k/epsilon(2) 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/epsilon))(b)/epsilon 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.