No Dense Subgraphs Appear in the Triangle-free Graph Process

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

Section: Dynamic Concentrationmentioning

confidence: 99%

“…We satisfy the variation equations (13), (14), (15), (16), (17), (18) by setting the error functions to have the form…”

Section: Dynamic Concentrationmentioning

confidence: 99%

A note on the random greedy independent set algorithm

Bennett

Bohman

2016

“…. 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%

The C_ℓ‐free process

Warnke

2012

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-...

“…These results suggest that, perhaps surprisingly, during this initial phase the graph produced by the H-free process is very similar to the uniform random graph with the same number of edges. The behaviour of the H-free process in later steps is not well understood, and so far only some preliminary results [12,29] are known. For example, in [29] it was shown that whp very dense subgraphs never appear in the H-free process.…”

mentioning

confidence: 99%

When does the K₄‐free process stop?

Warnke

2012

The K4‐free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K4. Let G be the random maximal K4‐free graph obtained at the end of the process. We show that for some positive constant C, with high probability as n→∞, the maximum degree in G is at most Cn3/5logn5. This resolves a conjecture of Bohman and Keevash for the K4‐free process and improves on previous bounds obtained by Bollobás and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has Θ(n8/5logn5) edges and is ‘nearly regular’, i.e., every vertex has degree Θ(n3/5logn5). This answers a question of Erdős, Suen and Winkler for the K4‐free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least Ω(n2/5(logn)4/5/loglogn), which matches an upper bound obtained by Bohman up to a factor of Θ(loglogn). Our analysis of the K4‐free process also yields a new result in Ramsey theory: for a special case of a well‐studied function introduced by Erdős and Rogers we slightly improve the best known upper bound.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 355‐397, 2014