Summary: Can a Graph Have Distinct Regular Partitions?
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 () and Lov´asz and Szegedy (,), from a
previously known variant of the regularity lemma due to Alon et al. . The proof also provides
a deterministic polynomial time algorithm for finding such partitions.
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
. For a set of vertices A V , we denote by E(A) the set of edges of the graph induced by A in