The Final Size of the<i>C</i><sub>4</sub>-Free Process

Picollelli, Michael E.

doi:10.1017/s0963548311000368

Cited by 22 publications

(32 citation statements)

References 19 publications

(60 reference statements)

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“…Up to the constant our upper bound is best possible, since the results of Bohman and Keevash imply that for some c > 0, whp the minimum degree is at least

c {(n \log n)}^{1 / (ℓ - 1)}

. The special case

ℓ = 4

was proved independently by Picollelli ; since this manuscript was submitted Picollelli has independently also proved the case

ℓ \geq 4

. So, combining our findings with , we not only verify the mentioned conjecture of Osthus and Taraz , but establish the following stronger result.…”

Section: Introductionsupporting

confidence: 75%

“…. In this work we consider a natural variant of the above process which has very recently received a considerable amount of attention .…”

Section: Introductionmentioning

confidence: 99%

“…After Bohman's result for C 3 , the next case to be resolved was

H = K_{4}

, for which matching bounds have been obtained by the author , and, independently, by Wolfovitz . During the preparation of this paper Picollelli also resolved the cases

H \in {C_{4}, K_{4}^{-}}

. But despite this progress, since the upper bound for the maximum degree d process in is immediate, one can argue that non‐trivial matching upper bounds have not been determined for any class of graphs.…”

Section: Introductionmentioning

confidence: 99%

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The C_ℓ‐free process

Warnke

2012

Random Struct Algorithms

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Abstract. The C ℓ -free process starts with the empty graph on n vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of C ℓ is created. For every ℓ ≥ 4 we show that, with high probability as n → ∞, the maximum degree is O((n log n) 1/(ℓ−1) ), which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the C ℓ -free process typically terminates with Θ(n ℓ/(ℓ−1) (log n) 1/(ℓ−1) ) edges, which answers a question of Erdős, Suen and Winkler. This is the first result that determines the final number of edges of the more general H-free process for a nontrivial class of graphs H. We also verify a conjecture of Osthus and Taraz concerning the average degree, and obtain a new lower bound on the independence number. Our proof combines the differential equation method with a tool that might be of independent interest: we establish a rigorous way to 'transfer' certain decreasing properties from the binomial random graph to the H-free process. IntroductionThe random graph process was introduced by Erdős and Rényi [10] in 1959. It starts with the empty graph on n vertices and adds new edges one by one, where each edge is chosen uniformly at random among all edges not yet present. Since then it has been studied extensively, and many tools and methods for investigating its typical properties have been developed, see e.g. [5,8,12]. In this work we consider a natural variant of the above process which has very recently received a considerable amount of attention [2,3,11,17,18,23,24,25,26,27].The H-free process was suggested by Bollobás and Erdős [4] in 1990, as a way to generate an interesting probability distribution on the set of maximal H-free graphs with potential applications to Ramsey Theory. Given some fixed graph H, it is a modification of the classical random graph process, where each new edge is chosen uniformly at random subject to the condition that no copy of H is formed. It was first described in print in 1995 by Erdős, Suen and Winkler [9], who asked how many edges the final graph typically has (this also appears as a problem in [7]). The main difficulty when analysing this process is that there is a complicated dependence among the edges; the order in which they are inserted is also relevant.The first results addressed certain special graphs, determining the typical final number of edges up to logarithmic factors. The case H = C 3 was studied in 1995 by Erdős, Suen and Winkler [9], and in 2000 Bollobás and Riordan [6] considered H ∈ {K 4 , C 4 }. In fact, a result of Ruciński and Wormald [21] predates those mentioned above: in 1992 they considered the (much simpler) maximum degree d-process, which corresponds to the case H = K 1,d+1 , and showed that whp 1 it ends with ⌊nd/2⌋ edges. The general H-free process was first analysed independently by Bollobás and Riordan [6] and Osthus and Taraz [16] in 2000. In fact, they assumed that H satisfies a certain density condition (strictly 2-...

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“…Up to the constant our upper bound is best possible, since the results of Bohman and Keevash imply that for some c > 0, whp the minimum degree is at least

c {(n \log n)}^{1 / (ℓ - 1)}

. The special case

ℓ = 4

was proved independently by Picollelli ; since this manuscript was submitted Picollelli has independently also proved the case

ℓ \geq 4

. So, combining our findings with , we not only verify the mentioned conjecture of Osthus and Taraz , but establish the following stronger result.…”

Section: Introductionsupporting

confidence: 75%

“…. In this work we consider a natural variant of the above process which has very recently received a considerable amount of attention .…”

Section: Introductionmentioning

confidence: 99%

“…After Bohman's result for C 3 , the next case to be resolved was

H = K_{4}

, for which matching bounds have been obtained by the author , and, independently, by Wolfovitz . During the preparation of this paper Picollelli also resolved the cases

H \in {C_{4}, K_{4}^{-}}

Section: Introductionmentioning

confidence: 99%

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The C_ℓ‐free process

Warnke

2012

Random Struct Algorithms

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“…Bohman and Keevash conjecture that this is the case for the H ‐free process when H is strictly 2‐balanced, but even this remains widely open. The conjecture has been verified in some special cases, including the K 3 ‐free process , the K 4 ‐free process and the

C_{scriptl}

‐free process for all

l \geq 4

.…”

Section: Introductionmentioning

confidence: 74%

“…It is tempting to speculate that the lower bound in Theorem 1.1 gives the correct order of magnitude of the maximal independent set produced by the random greedy independent set algorithm for a broad class of hypergraphs H. Bohman and Keevash conjecture that this is the case for the H-free process when H is strictly 2-balanced, but even this remains widely open. The conjecture has been verified in some special cases, including the K 3 -free process [3], the K 4 -free process [26,27] and the C -free process for all ≥ 4 [21,22,25].…”

Section: Introductionmentioning

confidence: 84%

A note on the random greedy independent set algorithm

Bennett

Bohman

2016

Random Struct Algorithms

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Let r be a fixed constant and let scriptH be an r‐uniform, D‐regular hypergraph on N vertices. Assume further that D>Nϵ for some ϵ>0. Consider the random greedy algorithm for forming an independent set in scriptH. An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly at random from the collection of vertices that could be added to the independent set (i.e. the collection of vertices v with the property that v is not in the current independent set I and I∪{v} contains no edge in scriptH). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of scriptH; that is, the process terminates at a maximal independent set. We prove that if scriptH satisfies certain degree and codegree conditions then there are Ωtrue(N·((logN)/D)1r−1true) vertices in the independent set produced by the random greedy algorithm with high probability. This result generalizes a lower bound on the number of steps in the H‐free process due to Bohman and Keevash and produces objects of interest in additive combinatorics. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 479–502, 2016

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

Makai

2014

Journal of Graph Theory

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Abstract:Consider the random graph process that starts from the complete graph on n vertices. In every step, the process selects an edge uniformly at random from the set of edges that are in a copy of a fixed graph H and removes it from the graph. The process stops when no more copies of H exist. When H is a strictly 2-balanced graph we give the exact asymptotics on the number of edges remaining in the graph when the process terminates and investigate some basic properties namely the size of the maximal independent set and the presence of subgraphs. C 2014 Wiley Periodicals, Inc. J. Graph Theory 79: [125][126][127][128][129][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144] 2015

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The Final Size of theC₄-Free Process

Cited by 22 publications

References 19 publications

The C_ℓ‐free process

The C_ℓ‐free process

A note on the random greedy independent set algorithm

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

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The Final Size of theC4-Free Process

Cited by 22 publications

References 19 publications

The Cℓ‐free process

The Cℓ‐free process

A note on the random greedy independent set algorithm

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

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The Final Size of theC₄-Free Process

The C_ℓ‐free process

The C_ℓ‐free process