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Testing Expansion in Bounded-Degree Graphs

Published online by Cambridge University Press:  09 June 2010

ARTUR CZUMAJ  and
CHRISTIAN SOHLER
ARTUR CZUMAJ
Affiliation:
Department of Computer Science and Centre for Discrete Mathematics and its Applications (DIMAP), University of Warwick, Coventry CV4 7AL, UK (e-mail: A.Czumaj@warwick.ac.uk)
CHRISTIAN SOHLER
Affiliation:
Department of Computer Science, Technische Universität Dortmund, D-44221 Dortmund, Germany (e-mail: christian.sohler@tu-dortmund.de)
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Abstract

We consider the problem of testing expansion in bounded-degree graphs. We focus on the notion of vertex expansion: an α-expander is a graph G = (V, E) in which every subset U ⊆ V of at most |V|/2 vertices has a neighbourhood of size at least α ⋅ |U|. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time . We prove that the property-testing algorithm proposed by Goldreich and Ron with appropriately set parameters accepts every α-expander with probability at least and rejects every graph that is ϵ-far from any α*-expander with probability at least , where and d is the maximum degree of the graphs. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is .

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