Can a Graph Have Distinct Regular Partitions?

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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 ([11]) and Lovász and Szegedy ([9,10]), from a previously known variant of the regularity lemma due to Alon et al.[2]. The proof also provides a deterministic polynomial time algorithm for finding such partitions.KeywordsBipartite GraphEdge DensityRegularity LemmaRegular PartitionSackler FacultyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.