Testing k-colorability

Abstract

Let G be a graph on n vertices and suppose that at least $\epsilon n^2$ 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 $c k\,{\rm ln}\,k/ \epsilon^2$ 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 $\frac{({\rm ln} (1/\epsilon))^b}{\epsilon}$ 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.