Can a Graph Have Distinct Regular Partitions?

The regularity lemma of Szemerédi gives a concise approximate description of a graph via a so-called regular partition of its vertex set. In this paper we address the following problem: Can a graph have two “distinct” regular partitions? It turns out that (as observed by several researchers) for the standard notion of a regular partition, one can construct a graph that has very distinct regular partitions. On the other hand, we show that for the stronger notion of a regular partition that has been recently studied, all such regular partitions of the same graph must be very “similar.” En route, we also give a short argument for deriving a recent variant of the regularity lemma obtained independently by Rödl and Schacht and by Lovász and Szegedy from a previously known variant of the regularity lemma due to Alon et al. in 2000. The proof also provides a deterministic polynomial time algorithm for finding such partitions.