Saturation of Berge Hypergraphs

English, Sean; Graber, Nathan; Kirkpatrick, Pamela; Methuku, Abhishek; Sullivan, Eric

doi:10.48550/arxiv.1710.03735

Cited by 4 publications

(9 citation statements)

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“…determined in [9], but not much is known about larger complete graphs. If ℓ ≥ k + 2, then it can be seen that sat k (n, K ℓ ) ≤ ℓ−1 k (n − ℓ + 2).…”

Section: Note That H[a] = K (K)mentioning

confidence: 99%

“…Determining the saturation numbers for Berge cycles would also be very interesting. Some upper bounds on cycles are given in [9], but no non-trivial lower bounds are known.…”

Section: Note That H[a] = K (K)mentioning

confidence: 99%

“…Similarly to the proof of Theorem 1.3 from [9], our main result involves finding a hypergraph with certain structural properties. A hypergraph H is linear if for every pair of edges e, f ∈ E(H), |e ∩ f |≤ 1.…”

Section: Introductionmentioning

confidence: 99%

“…In the seminal paper on saturation numbers for Berge hypergraphs, saturation number for Berge hypergraphs for many common classes of graphs were studied by the second author and others, including triangles, matchings, cycles, paths and stars [9]. 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].…”

Section: Introductionmentioning

confidence: 99%

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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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“…determined in [9], but not much is known about larger complete graphs. If ℓ ≥ k + 2, then it can be seen that sat k (n, K ℓ ) ≤ ℓ−1 k (n − ℓ + 2).…”

Section: Note That H[a] = K (K)mentioning

confidence: 99%

“…Determining the saturation numbers for Berge cycles would also be very interesting. Some upper bounds on cycles are given in [9], but no non-trivial lower bounds are known.…”

Section: Note That H[a] = K (K)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nearly-Regular Hypergraphs and Saturation of Berge Stars

Austhof

English

2019

Electron. J. Combin.

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“…In the same setting of k-uniform hypergraphs, English et al [2] proved that there are B k (G) saturated hypergraphs on n vertices and O(n) hyperedges, where B k (G) is the set of all k-uniform Berge-G hypergraphs, 3 ≤ k ≤ 5. See also English et al [3], for Berge saturation results on some special graphs.…”

Section: Introductionmentioning

confidence: 97%