Weak quasi‐randomness for uniform hypergraphs

ABSTRACT:The quasi-random theory for graphs mainly focuses on a large equivalent class of graph properties each of which can be used as a certificate for randomness. For k-graphs (i.e., kuniform hypergraphs), an analogous quasi-random class contains various equivalent graph properties including the k-discrepancy property (bounding the number of edges in the generalized induced subgraph determined by any given (k − 1)-graph on the same vertex set) as well as the k-deviation property (bounding the occurrences of "octahedron", a generalization of 4-cycle). In a 1990 paper (Chung, Random Struct Algorithms 1 (1990) 363-382), a weaker notion of l-discrepancy properties for k-graphs was introduced for forming a nested chain of quasi-random classes, but the proof for showing the equivalence of l-discrepancy and l-deviation, for 2 ≤ l < k, contains an error. An additional parameter is needed in the definition of discrepancy, because of the rich and complex structure in hypergraphs. In this note, we introduce the notion of (l, s)-discrepancy for k-graphs and prove that the equivalence of the (k, s)-discrepancy and the s-deviation for 1 ≤ s ≤ k. We remark that this refined notion of discrepancy seems to point to a lattice structure in relating various quasi-random classes for hypergraphs.

Section: Introductionmentioning

confidence: 83%

“…However, this inequality is not true for l = k as evidenced by the example given int [6]. So, a natural question is to find the 'right' equivalent discrepancy property for dev l .…”

Section: Namely E(h G) Counts the Number Of Ordered Subsets In E(h G)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quasi‐random hypergraphs revisited

Chung

2011

The quasi‐randomness of hypergraph cut properties

Shapira

Yuster

2011

Let α 1 , . . . , α k satisfy ∑ i α 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 , . . . , A k of sizes α 1 n, . . . , α k n, the number of edges intersecting A 1 , . . . , 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 While hypergraphs satisfying the property corresponding to α 1 = · · · = α 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.

The poset of hypergraph quasirandomness

Lenz

Mubayi

2013

Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-GrahamWilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them.Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's.