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An -vertex graph is called a small-set expander if any set of size ( ) contains at most a (1) fraction of the edges that touch it.The goal of this paper is to investigate small-set expansion properties of the Johnson graph, which is not a small-set expander.We obtain a qualitative descriptions of all small sets that violate the small-set expansion property in the Johnson graph: we show that any such set is correlated with some union of small intersections of "basic sets," where each basic set is a dictatorship-i.e., belonging to it depends only on containing a single coordinate .This condition is necessary and sufficient, since any such set violates the small-set expansion property.

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Publication Details

Journal
Theory of Computing
Published
2025-01-01
DOI
https://doi.org/10.4086/toc.2025.v021a002
Primary Topic
Limits and Structures in Graph Theory
Type
article
Field-Weighted Citation Impact
0.00

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Muli Safra, Dana Moshkovitz, Dor Minzer, Subhash Khot
Theory of Computing
Limits and Structures in Graph Theory
article

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Muli Safra, Dana Moshkovitz, Dor Minzer, Subhash Khot
article en

Abstract

An -vertex graph is called a small-set expander if any set of size ( ) contains at most a (1) fraction of the edges that touch it.The goal of this paper is to investigate small-set expansion properties of the Johnson graph, which is not a small-set expander.We obtain a qualitative descriptions of all small sets that violate the small-set expansion property in the Johnson graph: we show that any such set is correlated with some union of small intersections of "basic sets," where each basic set is a dictatorship-i.e., belonging to it depends only on containing a single coordinate .This condition is necessary and sufficient, since any such set violates the small-set expansion property.

Theory of ComputingVol. 21(1)
National Science Foundation, European Commission, Israel Science Foundation, Tel Aviv University
Openalex Percentile: Top 82%
Limits and Structures in Graph Theory
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