Tuza's conjecture for random graphs

Park, Jin‐Young

doi:10.1002/rsa.21057

Cited by 3 publications

(7 citation statements)

References 13 publications

(33 reference statements)

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“…Let r ≥ 3 and l ≤ r/2. Then, for any r-graph G, Finally, for d = Θ(1), extending ideas of Kahn-Park [12], we obtain the following.…”

Section: Results and Organizationmentioning

confidence: 76%

“…That is, δ(X, Y ) is half the ℓ 1 distance between the distributions of X and Y , and is the minimum of P (X = Y ) under couplings of X and Y . We require the following fact, which appears in [12]. Here Bin and Po are the Binomial and Poisson distributions, respectively.…”

Section: Probabilistic Toolsmentioning

confidence: 99%

“…Before proving (8), we need some notation and definitions. Note that the following proof is a straightforward generalization of the ideas in [12], and is based on the coupling from Section 3.…”

Section: Proof Of Theorem 111mentioning

confidence: 99%

“…The best known general bound is τ t (G) ≤ 66 23 ν t (G), due to Haxell [9]. In a recent development, Kahn and Park [12], and (independently) Bennett, Cushman and Dudek [4] show that Tuza's conjecture is true for the Erdős-Rényi random graph G(n, p).…”

Section: Introductionmentioning

confidence: 99%

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Generalized Tuza's conjecture for random hypergraphs

Basit¹,

Galvin²

2022

Preprint

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A celebrated conjecture of Tuza states that in any finite graph the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. For an r-uniform hypergraph (r-graph) G, let τ (G) be the minimum size of a cover of edges by (r − 1)-sets of vertices, and let ν(G) be the maximum size of a set of edges pairwise intersecting in fewer than r − 1 vertices. Aharoni and Zerbib proposed the following generalization of Tuza's conjecture:Let H r (n, p) be the uniformly random r-graph on n vertices. We show that, for r ∈ {3, 4, 5} and any p = p(n), H r (n, p) satisfies the Aharoni-Zerbib conjecture with high probability (i.e., with probability approaching 1 as n → ∞). We also show that there is a C < 1 such that, for any r ≥ 6 and any p = p(n), τ (H r (n, p))/ν(H r (n, p)) ≤ Cr with high probability. Furthermore, we may take C < 1/2 + ε, for any ε > 0, by restricting to sufficiently large r (depending on ε).

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“…Let r ≥ 3 and l ≤ r/2. Then, for any r-graph G, Finally, for d = Θ(1), extending ideas of Kahn-Park [12], we obtain the following.…”

Section: Results and Organizationmentioning

confidence: 76%

Section: Probabilistic Toolsmentioning

confidence: 99%

Section: Proof Of Theorem 111mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalized Tuza's conjecture for random hypergraphs

Basit¹,

Galvin²

2022

Preprint

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show abstract

“…for G(n, m) for any range of m. Theorem 1.4 will be proved in Section 3. We recently learned that Theorem 1.4 was independently proved by Kahn and Park [23] using a very different approach.…”

Section: Introductionmentioning

confidence: 99%

Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

Bennett¹,

Cushman²,

Dudek³

2020

Preprint

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A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number τ (G) is at most twice the triangle packing number ν(G), where the triangle packing number ν(G) is the maximum size of a set of edge-disjoint triangles in G and the triangle covering number τ (G) is the minimal size of a set of edges intersecting all triangles. In this paper, we prove that Tuza's conjecture holds in the Erdős-Rényi random graph G(n, m) for all range of m, closing the "gap" in what was previously known. (Recently, this result was also independently proved by Jeff Kahn and Jinyoung Park.) We employ a random greedy process called the online triangle packing process to produce a triangle packing in G(n, m) and analyze this process by using the differential equations method.

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Generalized Tuza’s Conjecture for Random Hypergraphs

Basit,

Galvin

2024

SIAM J. Discrete Math.

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Tuza's conjecture for random graphs

Cited by 3 publications

References 13 publications

Generalized Tuza's conjecture for random hypergraphs

Generalized Tuza's conjecture for random hypergraphs

Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

Generalized Tuza’s Conjecture for Random Hypergraphs

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