Summary: Testing k-colorability
Let G be a graph on n vertices and suppose that at least n2
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/ 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/ ))b
are non-bipartite, for some absolute positive constant
b, and this is tight up to the poly-logarithmic factor. Both results are motivated by the study
of testing algorithms for k-colorability, first considered by Goldreich, Goldwasser and Ron in ,
and improve the results in that paper.
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 k-colorable. Clearly G contains many non-k-
colorable subgraphs. Some of them are probably quite small in order. What is then the smallest
non-k-colorable subgraph of G? How many small non-k-colorable 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 non-k-colorable or alternatively -far from being k-colorable,