On the stability of graph independence number

Dong, Zichao; Wu, Zhuo

doi:10.48550/arxiv.2102.13306

Cited by 2 publications

(4 citation statements)

References 9 publications

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“…As we will see, in the case of regular graphs, the first example gives an almost tight bound for its parameter range. We remark that Theorem 4.3 settles the final open problem raised by Dong and Wu in [7].…”

Section: Blockers (Hitting Sets) In Graphssupporting

confidence: 56%

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The success probability in Levine's hat problem, and independent sets in graphs

Alon¹,

Friedgut²,

Kalai³

et al. 2022

Preprint

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Lionel Levine's hat challenge has t players, each with a (very large, or infinite) stack of hats on their head, each hat independently colored at random black or white. The players are allowed to coordinate before the random colors are chosen, but not after. Each player sees all hats except for those on her own head. They then proceed to simultaneously try and each pick a black hat from their respective stacks. They are proclaimed successful only if they are all correct. Levine's conjecture is that the success probability tends to zero when the number of players grows. We prove that this success probability is strictly decreasing in the number of players, and present some connections to problems in graph theory: relating the size of the largest independent set in a graph and in a random induced subgraph of it, and bounding the size of a set of vertices intersecting every maximum-size independent set in a graph.

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Section: Blockers (Hitting Sets) In Graphssupporting

confidence: 56%

“…We thank Wojtech Samotij and Bhargav Narayanan for useful discussions, and Zichao Dong and Zhuo Wu for telling us about [7].…”

Section: Acknowledgmentsmentioning

confidence: 99%

The success probability in Levine's hat problem, and independent sets in graphs

Alon¹,

Friedgut²,

Kalai³

et al. 2022

Preprint

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show abstract

“…On the other hand we observe that an old result of Hajnal implies that the assertion of the conjecture (with k = 1 and τ = 2α − 1) does hold for any α > 1/2. Theorem 1.4 also settles the final open problem raised by Dong and Wu in [6].…”

supporting

confidence: 65%

“…Acknowledgment I thank Ehud Friedgut for helpful discussions and Zichao Dong and Zhuo Wu for telling me about [6].…”

mentioning

confidence: 99%

Hitting all maximum independent sets

Alon

2021

Preprint

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We describe an infinite family of graphs G n , where G n has n vertices, independence number at least n/4, and no set of less than √ n/2 vertices intersects all its maximum independent sets. This is motivated by a question of Bollobás, Erdős and Tuza, and disproves a recent conjecture of Friedgut, Kalai and Kindler. Motivated by a related question of the last authors, we show that for every graph G on n vertices with independence number (1/4 + ε)n, the average independence number of an induced subgraph of G on a uniform random subset of the vertices is at most (1/4+ε−Ω(ε 2 ))n. Background and resultsThe following conjecture appears in a recent paper of Friedgut, Kalai and Kindler.Conjecture 1.1 ([8], Conjecture 3.1). For every α ∈ (0, 1/2) there exists k and τ > 0 such that if G is a graph on n vertices with maximum independent set of size αn, then there exist pairwise disjoint subsets of vertices A 1 , A 2 , . . . , A r in G such that

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On the stability of graph independence number

Cited by 2 publications

References 9 publications

The success probability in Levine's hat problem, and independent sets in graphs

The success probability in Levine's hat problem, and independent sets in graphs

Hitting all maximum independent sets

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