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The Quasi-randomness of Hypergraph Cut Properties by Asaf Shapira; Raphael Yuster is a Mathematics article available to read on EtoBox.

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## Abstract Let $\alpha\_1,\ldots,\alpha\_k$ satisfy $\sum\_i\alpha\_i=1$ and suppose a __k__‐uniform hypergraph on __n__ vertices satisfies the following property; in any partition of its vertices into __k__ sets $A\_1,\ldots,A\_k$ of sizes $\alpha\_1n,\ldots,\alpha\_kn$, the number of edges intersecting $A\_1,\ldots,A\_k$ is (asymptotically) the number one would expect to find in a random __k__‐uniform hypergraph. Can we then infer that __H__ is quasi‐random? We show that the answer is negative if and only if $\alpha\_1=\cdots=\alpha\_k=1/k$. This resolves an open problem raised in 1991 by Chung and Graham [J AMS 4 (1991), 151–196]. While hypergraphs satisfying the property corresponding to $\alpha\_1=\cdots=\alpha\_k=1/k$ are not necessarily quasi‐random, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasi‐random hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

Who reads The Quasi-randomness of Hypergraph Cut Properties?

Author

Asaf Shapira; Raphael Yuster

Publisher

John Wiley and Sons; Wiley (John Wiley & Sons); John Wiley & Sons Inc.; Wiley (ISSN 1042-9832)

Published

2011

Language

EN

Field

Mathematics (Physical Sciences)