A Note on Saturation for Berge-G Hypergraphs

Axenovich, Maria; Winter, Christian

doi:10.1007/s00373-019-02047-w

Cited by 8 publications

(10 citation statements)

References 16 publications

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“…For h ∈ {x, y}, let L(h) be the family of edges {h, u, v} such that u, v ∈ L \ R 2 , u is a 1-flat vertex of degree at most 8, and d(hv) = 1. We will show that |L(h)| ≤ 8 2 , which will imply that |R 3 | ≤ 16 2 .…”

Section: Lower Bound: Preliminary Lemmasmentioning

confidence: 85%

“…Suppose to the contrary that |L(h)| > 8 2 and let e = {h, u, v} ∈ L. Let A 1 be the set of the vertices in L ∩ N [{u, v}], and…”

Section: Lower Bound: Preliminary Lemmasmentioning

confidence: 99%

“…In terms of hypergraphs, most of the specific saturation numbers determined outside of complete graphs involve forbidden families of hypergraphs, such as triangular families [25], intersecting hypergraphs [8], and very recently Berge hypergraphs (e.g. [1,2,10,11,17]). For a detailed dynamic survey on all aspects of saturation in graphs and hypergraphs, see [14].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Saturation for the $3$-uniform loose $3$-cycle

English¹,

Kostochka²,

Zirlin³

2022

Preprint

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Let F and H be k-uniform hypergraphs. We say H is F -saturated if H does not contain a subgraph isomorphic to F , but H + e does for any hyperedge e ∈ E(H). The saturation number of F , denoted sat k (n, F ), is the minimum number of edges in a F -saturated k-uniform hypergraph H on n vertices. Let C(3) 3 denote the 3-uniform loose cycle on 3 edges. In this work, we prove that3 ) ≤This is the first non-trivial result on the saturation number for a fixed short hypergraph cycle.

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Section: Lower Bound: Preliminary Lemmasmentioning

confidence: 85%

“…Suppose to the contrary that |L(h)| > 8 2 and let e = {h, u, v} ∈ L. Let A 1 be the set of the vertices in L ∩ N [{u, v}], and…”

Section: Lower Bound: Preliminary Lemmasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Saturation for the $3$-uniform loose $3$-cycle

English¹,

Kostochka²,

Zirlin³

2022

Preprint

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show abstract

“…We say an r-uniform hypergraph H is a Berge-G if there exists a bijection φ : E(G) → E(H) such that e ⊆ φ(e) for each edge e ∈ E(G). Recently, extremal problems for Berge hypergraphs have attracted the attention of a lot of researchers, see, e.g., [3,2,4,5,8,13]. In 2018, Austhof and English [2] studied the saturation number of Berge stars.…”

Section: Introductionmentioning

confidence: 99%

Saturation Number of Berge Stars in Random Hypergraphs

Liu

Kang

2020

Electron. J. Combin.

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Let $G$ be a graph. We say an $r$-uniform hypergraph $H$ is a Berge-$G$ if there exists a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each $e\in E(G)$. Given a family of $r$-uniform hypergraphs $\mathcal{F}$ and an $r$-uniform hypergraph $H$, a spanning sub-hypergraph $H'$ of $H$ is $\mathcal{F}$-saturated in $H$ if $H'$ is $\mathcal{F}$-free, but adding any edge in $E(H)\backslash E(H')$ to $H'$ creates a copy of some $F\in\mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges in an $\mathcal{F}$-saturated spanning sub-hypergraph of $H$. In this paper, we asymptotically determine the saturation number of Berge stars in random $r$-uniform hypergraphs.

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“…The growth rate of saturation numbers also have been studied by the second author and others, and it has been determined that sat k (n, Berge-F ) = O(n) for 3 ≤ k ≤ 5 [8]. Recently, Axenovich and Winter have begun considering Berge saturation for non-uniform hypergraphs, showing that there are Berge-F -saturated non-uniform hypergraphs with |E(F )|−1 edges for all graphs F except stars, in which there are saturated examples with |E(F )| edges [1]. Here we will study the saturation number for Berge stars in the uniform case.…”

Section: Introductionmentioning

confidence: 99%

Nearly-Regular Hypergraphs and Saturation of Berge Stars

Austhof

English

2019

Electron. J. Combin.

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Given a graph G, we say a k-uniform hypergraph H on the same vertex set contains a Berge-G if there exists an injection φ : E(G) → E(H) such that e ⊆ φ(e) for each edge e ∈ E(G). A hypergraph H is Berge-G-saturated if H does not contain a Berge-G, but adding any edge to H creates a Berge-G. The saturation number for Berge-G, denoted sat k (n, Berge-G) is the least number of edges in a k-uniform hypergraph that is Berge-G-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we also prove the existence of nearly-regular k-uniform hypergraphs, or k-uniform hypergraphs in which every vertex has degree r or r − 1 for some r ∈ Z, and less than k vertices have degree r − 1.

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A Note on Saturation for Berge-G Hypergraphs

Cited by 8 publications

References 16 publications

Saturation for the $3$-uniform loose $3$-cycle

Saturation for the $3$-uniform loose $3$-cycle

Saturation Number of Berge Stars in Random Hypergraphs

Nearly-Regular Hypergraphs and Saturation of Berge Stars

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