On Ryser’s Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs

Király, Zoltán; Tóthmérész, Lilla

doi:10.37236/6448

Cited by 7 publications

(13 citation statements)

References 16 publications

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“…Bustamente and Stein [15], and Király and Tóthmérész [42] independently studied the following problem.…”

Section: Upper Boundsmentioning

confidence: 99%

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Generalizations and Strengthenings of Ryser's Conjecture

DeBiasio¹,

Kamel²,

McCourt³

et al. 2021

Electron. J. Combin.

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Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far-reaching generalization of König's theorem is only known to be true for $r\leq 3$, or when $\nu(H)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $\alpha(G)$, there exists at most $(r-1)\alpha(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.

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“…Bustamente and Stein [15], and Király and Tóthmérész [42] independently studied the following problem.…”

Section: Upper Boundsmentioning

confidence: 99%

“…Problem 102 ( [15], [42]). Let r and t be integers with r 2 and t 1 and let H be an r-partite hypergraph in which every pair of edges intersects in at least t vertices.…”

Section: Upper Boundsmentioning

confidence: 99%

Generalizations and Strengthenings of Ryser's Conjecture

DeBiasio¹,

Kamel²,

McCourt³

et al. 2021

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…We summarize our findings in the following table. 3 n\ e 11 12 As one can see, there are many candidates. However, checking their covering number, none of them was an extremal example.…”

Section: -Uniform Casementioning

confidence: 99%

“…Our covering number calculator found in 2 hours that all of these candidates have covering number at most 4. We listed all the other search results in the following table : To check (16,12) The 9 vertices of degree 4 and 12 vertices of degree 2 give rise to a (12|2 12 4 9 ) pairwise balanced design, and there is a unique such object 6 the dual of AG(2, 3). Therefore, there is a cover of size 4: consider 3 points on an affine line L and the intersection of the two lines parallel to L.…”

Section: The 5-uniform Casementioning

confidence: 99%

Intersecting and $2$-intersecting hypergraphs with maximal covering number: the Erdős-Lovász theme revisited

Barát

2020

Preprint

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Erdős and Lovász noticed that an r-uniform intersecting hypergraph H with maximal covering number, that is τ (H) = r, must have at least 8 3 r − 3 edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let q(r) denote the minimum number of edges in an intersecting r-uniform hypergraph. It was known that q(3) = 6 and q(4) = 9. We obtain the following new results: The extremal example for uniformity 4 is unique. Somewhat surprisingly it is not symmetric by any means. For uniformity 5, q(5) = 13, and we found 3 examples, none of them being some known graph. We use both theoretical arguments and computer searches. In the footsteps of Erdős and Lovász, we also consider the special case, when the hypergraph is part of a finite projective plane. We determine the exact answer for r ∈ {3, 4, 5, 6}. For uniformity 6, there is a unique extremal example.In a related question, we try to find 2-intersecting r-uniform hypergraphs with maximal covering number, that is τ (H) = r − 1. An infinite family of examples is to take all possible r-sets of a (2r − 2)vertex set. There is also a geometric candidate: biplanes. These are symmetric 2-designs with λ = 2. We determined that only 3 biplanes of the 18 known examples are extremal.

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“…A hypergraph is t-intersecting if any two edges have at least t common vertices. Recently, Király and Tóthmérész [12] and independently Bustamante and Stein [3] conjectured the following t-intersecting generalization of Ryser's conjecture: If H is an r-partite hypergraph and any two hyperedges intersect in at least t vertices, then τ H r t ( ) − ≤ . Bishnoi et al [2] proved the tight upper bound r t ( − ) 2 + 1 ⌊ ∕ ⌋ for the range t 2 ≥ and r t 3 − 1 ≤…”

Section: Introductionmentioning

confidence: 99%

Intersecting and 2‐intersecting hypergraphs with maximal covering number: The Erdős–Lovász theme revisited

Barát

2020

J of Combinatorial Designs

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Erdős and Lovász noticed that an ‐uniform intersecting hypergraph with maximal covering number, that is, , must have at least edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let denote the minimum number of edges in an intersecting ‐uniform hypergraph. It was known that and . We obtain the following new results: The extremal example for uniformity 4 is unique. Somewhat surprisingly it is not symmetric by any means. For uniformity 5, , and we found three examples, none of them being some known graph. We use both theoretical arguments and computer searches. In the footsteps of Erdős and Lovász, we also consider the special case, when the hypergraph is part of a finite projective plane. We determine the exact answer for . For uniformity 6, there is a unique extremal example. In a related question, we try to find 2‐intersecting r‐uniform hypergraphs with maximal covering number, that is, . An infinite family of examples is to take all possible r‐sets of a ‐vertex set. There is also a geometric candidate: biplanes. These are symmetric 2‐designs with . We determined that only three biplanes of the 18 known examples are extremal.

show abstract

On Ryser’s Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs

Cited by 7 publications

References 16 publications

Generalizations and Strengthenings of Ryser's Conjecture

Generalizations and Strengthenings of Ryser's Conjecture

Intersecting and $2$-intersecting hypergraphs with maximal covering number: the Erdős-Lovász theme revisited

Intersecting and 2‐intersecting hypergraphs with maximal covering number: The Erdős–Lovász theme revisited

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