Holes in Graphs

Yuejian Peng, Vojtech Rödl, Andrzej Ruciński
2001 Electronic Journal of Combinatorics  
The celebrated Regularity Lemma of Szemerédi asserts that every sufficiently large graph $G$ can be partitioned in such a way that most pairs of the partition sets span $\epsilon$-regular subgraphs. In applications, however, the graph $G$ has to be dense and the partition sets are typically very small. If only one $\epsilon$-regular pair is needed, a much bigger one can be found, even if the original graph is sparse. In this paper we show that every graph with density $d$ contains a large,
more » ... ively dense $\epsilon$-regular pair. We mainly focus on a related concept of an $(\epsilon,\sigma)$-dense pair, for which our bound is, up to a constant, best possible.
doi:10.37236/1618 fatcat:5gwqxcvue5bdhm6pikzrhtaxsi