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Can a Graph Have Distinct Regular Partitions? Asaf Shapira

Summary: Can a Graph Have Distinct Regular Partitions?
Noga Alon
Asaf Shapira
Uri Stav§
The regularity lemma of Szemer´edi 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 R¨odl and Schacht ([11]) and Lov´asz 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.
1 Introduction
We start with some of the basic definitions of regularity and state the regularity lemmas that we
refer to in this paper. For a comprehensive survey on the regularity lemma the reader is referred to
[7]. For a set of vertices A V , we denote by E(A) the set of edges of the graph induced by A in


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics