Hypergraphs with no cycle of length 4

Gyri, Ervin; Lemons, Nathan

doi:10.1016/j.disc.2011.06.027

Cited by 12 publications

(11 citation statements)

References 4 publications

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“…This improves the results of Győri and Lemons [8] who showed that the leading term of the sum above is between 1 8 n 3/2 and 12 √ 2n 3/2 . If a hypergraph H is C i -free for i = 2, 3, .…”

Section: Introductionsupporting

confidence: 88%

Extremal Results for Berge Hypergraphs

Gerbner¹,

Palmer²

2017

SIAM J. Discrete Math.

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Let G be a graph and H be a hypergraph both on the same vertex set. We say that a hypergraph H is a Berge-G if there is a bijection f : E(G) → E(H) such that for e ∈ E(G) we have e ⊂ f (e). This generalizes the established definitions of "Berge path" and "Berge cycle" to general graphs. For a fixed graph G we examine the maximum possible size (i.e. the sum of the cardinality of each edge) of a hypergraph with no Berge-G as a subhypergraph. In the present paper we prove general bounds for this maximum when G is an arbitrary graph. We also consider the specific case when G is a complete bipartite graph and prove an analogue of the Kővári-Sós-Turán theorem. *

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Section: Introductionsupporting

confidence: 88%

Extremal Results for Berge Hypergraphs

Gerbner¹,

Palmer²

2017

SIAM J. Discrete Math.

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“…Next, we consider the number of short cycles in H m (n, α). Interestingly, a cycle in a hypergraph can be defined in various ways ( [19,13,11,17,26,23]) and different types of cycle could contain different information about the global structure of a hypergraph ( [29]). In this paper, we focus on loose 2-cycle, the shortest cycle in H m (n, α) for m ≥ 3.…”

Section: Model and Main Resultsmentioning

confidence: 99%

Phase transition in a power-law uniform hypergraph

Yuan¹

2021

Preprint

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We propose a power-law m-uniform random hypergraph on n vertexes. In this hypergraph, each vertex is independently assigned a random weight from a power-law distribution with exponent α ∈ (0, ∞) and the hyperedge probabilities are defined as functions of the random weights. We characterize the number of hyperedge and the number of loose 2-cycle. There is a phase transition phenomenon for the number of hyperedge at α = 1. Interestingly, for the number of loose 2-cycle, phase transition occurs at both α = 1 and α = 2.

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“…Interestingly, they relate the question of estimating the maximum number of edges in a hypergraph of given girth with the famous question of estimating generalized Turán numbers initiated by Brown, Erdős and Sós [8] and show that the two problems are equivalent in some cases. Since then Turán-type extremal problems for hypergraphs in the Berge sense have attracted considerable attention: see e.g., [6,9,15,17,18,19,20,21,23,34].…”

Section: Introductionmentioning

confidence: 99%