Summary: A Combinatorial Characterization of the Testable Graph Properties:
It's All About Regularity
Extended Abstract + Appendix
A common thread in all the recent results concerning testing dense graphs is the use of Sze-
mer´edi's regularity lemma. In this paper we show that in some sense this is not a coincidence.
Our first result is that the property defined by having any given Szemer´edi-partition is testable
with a constant number of queries. Our second and main result is a purely combinatorial char-
acterization of the graph properties that are testable with a constant number of queries. This
characterization (roughly) says that a graph property P can be tested with a constant number of
queries if and only if testing P can be reduced to testing the property of satisfying one of finitely
many Szemer´edi-partitions. This means that in some sense, testing for Szemer´edi-partitions is as
hard as testing any testable graph property. We thus resolve one of the main open problems in
the area of property-testing, which was first raised in the 1996 paper of Goldreich, Goldwasser
and Ron  that initiated the study of graph property-testing. This characterization also gives
an intuitive explanation as to what makes a graph property testable.