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QUASI-RANDOMNESS AND ALGORITHMIC REGULARITY FOR GRAPHS WITH GENERAL DEGREE DISTRIBUTIONS
 

Summary: QUASI-RANDOMNESS AND ALGORITHMIC REGULARITY FOR
GRAPHS WITH GENERAL DEGREE DISTRIBUTIONS
NOGA ALON, AMIN COJA-OGHLAN, HI^E. P H`AN§, MIHYUN KANG¶, VOJTECH
RšODL , AND MATHIAS SCHACHT
Abstract. We deal with two intimately related subjects: quasi-randomness and regular par-
titions. The purpose of the concept of quasi-randomness is to measure how much a given graph
"resembles" a random one. Moreover, a regular partition approximates a given graph by a bounded
number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral charac-
terization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we
introduce a concept of regularity that takes into account vertex weights, and show that if G = (V, E)
satisfies a certain boundedness condition, then G admits a regular partition. In addition, building
on the work of Alon and Naor [Proc. 36th ACM STOC (2004) 72­80], we provide an algorithm
that computes a regular partition of a given (possibly sparse) graph G in polynomial time. As an
application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs
without "dense spots".
Key words: quasi-random graphs, Laplacian eigenvalues, regularity lemma, Grothendieck's inequal-
ity.
AMS subject classification: 05C85, 05C50.
1. Introduction and Results. This paper deals with quasi-randomness and
regular partitions. Loosely speaking, a graph is quasi-random if the global distribution

  

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

 

Collections: Mathematics