 
Summary: Testing kcolorability
Noga Alon
Michael Krivelevich
Abstract
Let G be a graph on n vertices and suppose that at least n2
edges have to be deleted from
it to make it kcolorable. It is shown that in this case most induced subgraphs of G on ck ln k/ 2
vertices are not kcolorable, where c > 0 is an absolute constant. If G is as above for k = 2,
then most induced subgraphs on (ln(1/ ))b
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 kcolorability, first considered by Goldreich, Goldwasser and Ron in [3],
and improve the results in that paper.
1 Introduction
Suppose that for a fixed integer k and a small > 0, a graph G = (V, E) on n vertices is such
that at least n2 edges should be deleted to make G kcolorable. Clearly G contains many nonk
colorable subgraphs. Some of them are probably quite small in order. What is then the smallest
nonkcolorable subgraph of G? How many small nonkcolorable subgraphs are there?
In order to address the above questions quantitatively, we introduce a suitable notation. First, we
call a graph G on n vertices robustly nonkcolorable or alternatively far from being kcolorable,
