The diamond‐free process

Picollelli, Michael E.

doi:10.1002/rsa.20517

Cited by 14 publications

(17 citation statements)

References 21 publications

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“…For this process Picollelli [20] shows that the number of steps is larger than the bound given by (2) by a logarithmic factor. Bennett [2] has recent results on the sum-free process.…”

Section: Introductionmentioning

confidence: 84%

“…When H is a diamond then the hypergraph

ℋ_{H}

is 5‐uniform and

5 (n - 2) (n - 3) / 2

‐regular but has

Δ_{3} (ℋ_{H}) = 3 (n - 3) = Θ (D^{1 / 2})

. For this process Picollelli shows that the number of steps is larger than the bound given by (2) by a logarithmic factor. Bennett has recent results on the sum‐free process.…”

Section: Introductionmentioning

confidence: 97%

“…A diamond is the graph obtained by removing an edge from K 4 . The diamond‐free process studied by Picollelli is an example of an H ‐free process where the graph H is 2‐balanced but not strictly 2‐balanced. When H is a diamond then the hypergraph

ℋ_{H}

is 5‐uniform and

5 (n - 2) (n - 3) / 2

‐regular but has

Δ_{3} (ℋ_{H}) = 3 (n - 3) = Θ (D^{1 / 2})

.…”

Section: Introductionmentioning

confidence: 99%

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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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“…For this process Picollelli [20] shows that the number of steps is larger than the bound given by (2) by a logarithmic factor. Bennett [2] has recent results on the sum-free process.…”

Section: Introductionmentioning

confidence: 84%

“…When H is a diamond then the hypergraph

ℋ_{H}

is 5‐uniform and

5 (n - 2) (n - 3) / 2

‐regular but has

Δ_{3} (ℋ_{H}) = 3 (n - 3) = Θ (D^{1 / 2})

. For this process Picollelli shows that the number of steps is larger than the bound given by (2) by a logarithmic factor. Bennett has recent results on the sum‐free process.…”

Section: Introductionmentioning

confidence: 97%

ℋ_{H}

is 5‐uniform and

5 (n - 2) (n - 3) / 2

‐regular but has

Δ_{3} (ℋ_{H}) = 3 (n - 3) = Θ (D^{1 / 2})

.…”

Section: Introductionmentioning

confidence: 99%

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A note on the random greedy independent set algorithm

Bennett

Bohman

2016

Random Struct Algorithms

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“…We believe that variants of Theorems 1 and 4 also hold for many other forbidden graphs (using semirandom variants of the H-free process [25,6,34,35,27]); we hope to return to this topic in a future work.…”

Section: Main Tool: Pseudo-random Triangle-free Subgraphsmentioning

confidence: 98%

“…where |X uv (i)| and |Z uv (i)| intuitively correspond to an 'open codegree' and the usual codegree, respectively (note that |Y uv (i)| corresponds to a 'mixed codegree', see (16)). Guided by Section 2.2, we define Ψ(x) as the unique solution to the differential equation Ψ ′ (x) = exp(−Ψ 2 (x)) and Ψ(0) = 0 from (27). With the heuristics (22) in mind, we introduce the parameters…”

Section: Definitions and Parametersmentioning

confidence: 99%

Packing Nearly Optimal Ramsey R(3,t) Graphs

Guo

Warnke

2020

Combinatorica

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In 1995 Kim famously proved the Ramsey bound R(3, t) ≥ ct 2 / log t by constructing an n-vertex graph that is triangle-free and has independence number at most C √ n log n. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph Kn into a packing of such nearly optimal Ramsey R(3, t) graphs. More precisely, for any ǫ > 0 we find an edge-disjoint collection (Gi)i of n-vertex graphs Gi ⊆ Kn such that (a) each Gi is triangle-free and has independence number at most Cǫ √ n log n, and (b) the union of all the Gi contains at least (1 − ǫ) n 2 edges. Our algorithmic proof proceeds by sequentially choosing the graphs Gi via a semi-random (i.e., Rödl nibble type) variation of the triangle-free process.As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabó (concerning a Ramsey-type parameter introduced by Burr, Erdős, Lovász in 1976). Namely, denoting by sr(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H = K3 and establish that sr(K3) = Θ(r 2 log r). The triangle-free process (proposed by Bollobás and Erdős) proceeds as follows: starting with an empty n-vertex graph, in each step a single edge is added, chosen uniformly at random from all non-edges which do not create a triangle.2 Kim's semi-random variation proceeds similar to the triangle-free process, but intuitively adds a large number of carefully chosen random-like edges in each step (instead of just a single edge); see Section 2 for more details. Main result: packing of nearly optimal Ramsey R(3, t) graphsKim and Bohman both proved the Ramsey bound R(3, t) = Ω(t 2 / log t) by showing the existence of a trianglefree graph G ⊆ K n on n vertices with independence number α(G) = O( √ n log n), which is best possible up to the value of the implicit constants. Our first theorem naturally extends their celebrated results, by approximately decomposing the complete graph K n into a packing of such nearly optimal Ramsey R(3, t) graphs.Theorem 1. For any ǫ > 0 there exist n 0 , C, D > 0 such that, for all n ≥ n 0 , there is an edge-disjointOur algorithmic proof proceeds by sequentially choosing the |I| = Θ( n/ log n) edge-disjoint triangle-free subgraphs√ n log n) via a semi-random variation of the triangle-free process akin Kim [20] (see Sections 1.3 and 2 for the details). In particular, we do not only show existence of the (G i ) i∈I , but also obtain a polynomial-time randomized algorithm which constructs these subgraphs. Theorem 1 improves a construction of Fox et.al. [16, Lemma 4.2], who used the basic Lovász Local Lemma based R(3, t)-approach to sequentially choose Θ( √ n/ log n) edge-disjoint triangle-free subgraphs with α(G i ) = O( √ n log n). It is natural to suspect that applying a more sophisticated R(3, t)-approach in each iteration ought to give an improved packing (with smaller independence number than the LLL approach), and here the usage of the triangle-free process was...

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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 diamond‐free process

Cited by 14 publications

References 21 publications

A note on the random greedy independent set algorithm

A note on the random greedy independent set algorithm

Packing Nearly Optimal Ramsey R(3,t) Graphs

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

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