The Reverse <i>H</i>‐free Process for Strictly 2‐Balanced Graphs

Makai, Tamás

doi:10.1002/jgt.21821

Cited by 7 publications

(13 citation statements)

References 29 publications

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“…(41) Proof of Theorem 15. We modify the proof of Theorem 2, where independence is only used to establish (27). Using (41) and that the bijection ρ k : Σ a → Σ b satisfies (40), we obtain…”

Section: Variants Using the General Lipschitz Conditionmentioning

confidence: 99%

“…Conditioned on B e = q, the decision whether e is added only depends on the edges f with B f ≤ q, which have the same distribution as G n,q . As noted by Makai [27], this allows for the use of classical random graph theory when estimating the probability that an edge is added to the evolving graph.…”

Section: Final Number Of Edges In the Reverse H-free Processmentioning

confidence: 99%

“…To establish Theorem 19 it remains to bound the expected final number of edges up to constant factors. Our argument is inspired by Makai [27], who proved asymptomatically matching bounds in (46) for the class of strictly 2-balanced graphs (the case H = K 3 is due to Erdős, Suen and Winkler [16]). In fact, here we determine the correct order of magnitude for all graphs.…”

Section: Lemma 26mentioning

confidence: 99%

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On the Method of Typical Bounded Differences

Warnke¹

2015

Combinator. Probab. Comp.

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Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that random functions are near their means. Of particular importance is the case where f (X) is a function of independent random variables X = (X1, . . . , Xn). Here the well known bounded differences inequality (also called McDiarmid's or Hoeffding-Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |fWhile this is easy to check, the main disadvantage is that it considers worst-case changes c k , which often makes the resulting bounds too weak to be useful.In this paper we prove a variant of the bounded differences inequality which can be used to establish concentration of functions f (X) where (i) the typical changes are small although (ii) the worst case changes might be very large. One key aspect of this inequality is that it relies on a simple condition that (a) is easy to check and (b) coincides with heuristic considerations why concentration should hold. Indeed, given an event Γ that holds with very high probability, we essentially relax the Lipschitz condition (L) to situations where Γ occurs. The point is that the resulting typical changes c k are often much smaller than the worst case ones.To illustrate its application we consider the reverse H-free process, where H is 2-balanced. We prove that the final number of edges in this process is concentrated, and also determine its likely value up to constant factors. This answers a question of Bollobás and Erdős.

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Section: Variants Using the General Lipschitz Conditionmentioning

confidence: 99%

Section: Final Number Of Edges In the Reverse H-free Processmentioning

confidence: 99%

Section: Lemma 26mentioning

confidence: 99%

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On the Method of Typical Bounded Differences

Warnke¹

2015

Combinator. Probab. Comp.

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“…Erdős, Suen and Winkler [12] proved that the expected number of edges in the final graph is (1 + o(1)) √ πn 3/2 /4. Makai [20] and independently Warnke [22] then proved that the final number of edges is concentrated about its expectation.…”

Section: Introductionmentioning

confidence: 96%

Large triangle packings and Tuza’s conjecture in sparse random graphs

Bennett

Dudek

Zerbib

2020

Combinator. Probab. Comp.

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The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

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“…We will describe this process more formally using an equivalent definition based on random permutations which turns out to be easier to work with. The approach was first used (in the K n host graph setting) by Erdős, Suen and Winkler [11] and has also appeared in a more refined form, for instance in [19,26]. We first choose a uniformly random permutation of E(Q d ) giving a labelling of these edges as e 1 , e 2 , .…”

Section: Introductionmentioning

confidence: 99%

The $Q_2$-Free Process in the Hypercube

Johnson

Pinto

2020

Electron. J. Combin.

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The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the $Q_2$-free process in $Q_d$ and the random subgraph of $Q_d$ it generates. Our main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.

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The Reverse H‐free Process for Strictly 2‐Balanced Graphs

Cited by 7 publications

References 29 publications

On the Method of Typical Bounded Differences

On the Method of Typical Bounded Differences

Large triangle packings and Tuza’s conjecture in sparse random graphs

The $Q_2$-Free Process in the Hypercube

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