 
Summary: A Combinatorial Characterization of the Testable Graph Properties:
It's All About Regularity
Extended Abstract + Appendix
Noga Alon
Eldar Fischer
Ilan Newman
Asaf Shapira§
Abstract
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´edipartition 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´edipartitions. This means that in some sense, testing for Szemer´edipartitions is as
hard as testing any testable graph property. We thus resolve one of the main open problems in
the area of propertytesting, which was first raised in the 1996 paper of Goldreich, Goldwasser
and Ron [24] that initiated the study of graph propertytesting. This characterization also gives
an intuitive explanation as to what makes a graph property testable.
