Linear hypergraphs with large transversal number and maximum degree two

Dorfling, Michael; Henning, Michael A.

doi:10.1016/j.ejc.2013.07.016

Cited by 28 publications

(24 citation statements)

References 14 publications

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“…As a consequence of results due to Tuza [39], Chvátal and McDiarmid [10], Henning and Yeo [20,22], and Dorfling and Henning [12], we have the following result. For k ≥ 2, let E k denote the k-uniform hypergraph on k vertices with exactly one edge.…”

Section: Motivationsupporting

confidence: 61%

“…For k ≥ 2, let E k denote the k-uniform hypergraph on k vertices with exactly one edge. Theorem 2 ([10,12,20,39]) For k ∈ {2, 3} if H is a k-uniform linear connected hypergraph, then τ (H) ≤ n + m k + 1 with equality if and only if H consists of a single edge or H is obtained from the affine plane AG (2, k) of order k by deleting one or two vertices; that is, H = E k or H ∈ {F k 2 −2 , F k 2 −1 }.…”

Section: Motivationmentioning

confidence: 99%

“…45α ′ (G) (11) = 32|S| + 8n 1 + 22n 2 + 42n 3 + 68n 4 + 88n 2 5 + 94n 1 5 + i∈Z Z 6 (20i − 6)n i +13|S| − 8n 1 + 23n 2 + 3n 3 + 22n 4 + 2n 2 5 − 4n 1 5 + Σ even + Σ odd (12) ≥ (8n + 6m + 6|Q|) + (13|S| − 8n 1 − 4n 1 5 )…”

Section: Claim L1: For Any Special Hunclassified

“…A subset T of vertices in a hypergraph H is a transversal (also called vertex cover or hitting set in many papers) if T has a nonempty intersection with every edge of H. The transversal number τ (H) of H is the minimum size of a transversal in H. A transversal of size τ (H) is called a τ (H)-transversal. Transversals in hypergraphs are well studied in the literature (see, for example, [7,10,11,12,18,19,20,21,22,23,31,32,37,39]).…”

Section: Introductionmentioning

confidence: 99%

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Transversals in 4-Uniform Hypergraphs

Henning

Yeo

2016

Electron. J. Combin.

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The transversal number τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. A k-uniform hypergraph has all edges of size k. Very few papers give bounds on the transversal number for linear hypergraphs, even though these appear in many applications, as it seems difficult to utilise the linearity in the known techniques. This paper is one of the first that give strong non-trivial bounds on the transversal number for linear hypergraphs, which is better than for non-linear hypergraphs. It is known that τ (H) ≤ (n + m)/(k + 1) holds for all k-uniform, linear hypergraphs H when k ∈ {2, 3} or when k ≥ 4 and the maximum degree of H is at most two. It has been conjectured (at several conference talks) that τ (H) ≤ (n + m)/(k + 1) holds for all k-uniform, linear hypergraphs H. We disprove the conjecture for large k, and show that the best possible constant c k in the bound τ (H) ≤ c k (n + m) has order ln(k)/k for both linear (which we show in this paper) and non-linear hypergraphs. We show that for those k where the conjecture holds, it is tight for a large number of densities if there exists an affine plane AG(2, k) of order k ≥ 2. We raise the problem to find the smallest value, k min , of k for which the conjecture fails. We prove a general result, which when applied to a projective plane of order 331 shows that k min ≤ 166. Even though the conjecture fails for large k, our main result is that it still holds for k = 4, implying that k min ≥ 5. The case k = 4 is much more difficult than the cases k ∈ {2, 3}, as the conjecture does not hold for general (non-linear) hypergraphs when k = 4. Key to our proof is the completely new technique of the deficiency of a hypergraph introduced in this paper.Claim H.6: The case |X| ≥ 2 and |∂(X)| ≥ 4 cannot occur.Proof of Claim H.6: Note that 8|X 4 |+5|X 14 |+4|X 11 |+|X 21 | ≤ 8|X| ≤ 13|X|−6|X ′ |−10, as |X| ≥ 2 and |X ′ | = 0. If H ′ is linear, then we are obtain a contradiction to Claim H.4(a), and if H ′ is not linear we obtain a contradiction to Claim H.5. (✷) Claim H.7: The case |X| ≥ 3 and |∂(X)| = 3 cannot occur.Proof of Claim H.7: Note that 8|X 4 | + 5|X 14 | + 4|X 11 | + |X 21 | ≤ 8|X| ≤ 13|X| − 6|X ′ | − 9, as |X| ≥ 3 and |X ′ | = 1. If H ′ is linear, then we are obtain a contradiction to Claim H.4(b), and if H ′ is not linear we obtain a contradiction to Claim H.5. (✷) Claim H.8: The case |X| = 2 and |∂(X)| = 3 cannot occur.Proof of Claim H.8: Suppose, to the contrary, that |X| = 2 and |∂(X)| = 3. Then, |E * (X)| = 3 and |X ′ | = 1. If H ′ is not linear, then we obtain a contradiction by Claim H.5(a), as 8|X 4 | + 5|X 14 | + 4|X 11 | + |X 21 | ≤ 16 and 13|X| − 6|X ′ | − 3 = 17. Therefore, H ′ is linear.

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Section: Motivationsupporting

confidence: 61%

Section: Motivationmentioning

confidence: 99%

Section: Claim L1: For Any Special Hunclassified

Section: Introductionmentioning

confidence: 99%

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Transversals in 4-Uniform Hypergraphs

Henning

Yeo

2016

Electron. J. Combin.

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“…Two of the five chapters in the major monograph of hypergraph theory [1] deal with transversals and their fractional version (real relaxation via Linear Programming). We refer to [6,7,13,21,22,23] for recent results and further references.…”

Section: Introductionmentioning

confidence: 99%

Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

Bujtás¹,

Henning²,

Tuza³

2016

SIAM J. Discrete Math.

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The 3 4 -Game Total Domination Conjecture posed by Henning, Klavžar and Rall [Combinatorica, to appear] states that if G is a graph on n vertices in which every component contains at least three vertices, then γ tg (G) ≤ 3 4 n, where γ tg (G) denotes the game total domination number of G. Motivated by this conjecture, we raise the problem to a higher level by introducing a transversal game in hypergraphs. We define the game transversal number, τ g (H), of a hypergraph H, and prove that if every edge of H has size at least 2, and H C 4 , then τ g (H) ≤ 4 11 (n H + m H ), where n H and m H denote the number of vertices and edges, respectively, in H. Further, we characterize the hypergraphs achieving equality in this bound. As an application of this result, we prove that if G is a graph on n vertices with minimum degree at least 2, then γ tg (G) < 8 11 n. As a consequence of this result, the 3 4 -Game Total Domination Conjecture is true over the class of graphs with minimum degree at least 2.

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Domination and Total Domination in Hypergraphs

Henning

Yeo

2020

Developments in Mathematics

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Linear hypergraphs with large transversal number and maximum degree two

Cited by 28 publications

References 14 publications

Transversals in 4-Uniform Hypergraphs

Transversals in 4-Uniform Hypergraphs

Transversal Game on Hypergraphs and the $\frac{3}{4}$-Conjecture on the Total Domination Game

Domination and Total Domination in Hypergraphs

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