Random maximal H‐free graphs

Osthus, Deryk; Taraz, Anusch

doi:10.1002/1098-2418(200101)18:1<61::aid-rsa5>3.3.co;2-k

Cited by 37 publications

(80 citation statements)

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“…A random maximal P-graph is obtained from n isolated vertices by randomly adding those edges (at each stage choosing uniformly among edges whose inclusion would not destroy property P ) until no further edges can be added. The question of finding the number of edges of a random maximal P -graph for several properties P is well studied [33,16,38,8,27]. In particular, when P is the property that the graph has girth greater than k, [27] shows that the above process of sequentially growing the graph leads to graphs with m = O(n 1+ 1 k−1 log n) edges.…”

Section: Introductionmentioning

confidence: 99%

“…The question of finding the number of edges of a random maximal P -graph for several properties P is well studied [33,16,38,8,27]. In particular, when P is the property that the graph has girth greater than k, [27] shows that the above process of sequentially growing the graph leads to graphs with m = O(n 1+ 1 k−1 log n) edges. Unfortunately, these random maximal P -graphs may have distribution that are far from uniform.…”

Section: Introductionmentioning

confidence: 99%

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Generating Random Graphs with Large Girth

Bayati

Montanari

Saberi

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present a simple and efficient algorithm for randomly generating simple graphs without small cycles. These graphs can be used to design high performance Low-Density Parity-Check (LDPC) codes. For any constant k, α ≤ 1/2k(k + 3) and m = O(n 1+α ), our algorithm generates an asymptotically uniform random graph with n vertices, m edges, and girth larger than k in polynomial time. To the best of our knowledge this is the first polynomial algorithm for the problem.Our algorithm generates a graph by sequentially adding m edges to an empty graph with n vertices. Recently, this type of sequential process has been very successful for efficiently counting and generating random graphs [35,18,11,7,5,6].

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generating Random Graphs with Large Girth

Bayati

Montanari

Saberi

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…For example, it is natural to ask about the k ‐path‐free process, or the restricted‐girth process where we greedily add edges keeping the girth above some value k . (The restricted‐girth process has already been studied for fixed k by Osthus and Taraz ; see also the work of Bayati, Montanari and Saberi on a slightly different process.) There are also natural generalizations of these processes to hypergraphs.…”

Section: Discussionmentioning

confidence: 98%

The random k‐matching‐free process

Krivelevich

Kwan²,

Loh

et al. 2018

Random Struct Algorithms

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Let scriptP be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty n‐vertex graph and then adds edges one‐by‐one, each chosen uniformly at random subject to the constraint that scriptP is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the k‐matching‐free process where scriptP is the property of not containing a matching of size k. We are able to analyze the behavior of this process for a wide range of values of k; in particular we prove that if k = o(n) or if n−2k=ofalse(nfalse/normallognfalse) then this process is likely to terminate in a k‐matching‐free graph with the maximum possible number of edges, as characterized by Erdős and Gallai. We also show that these bounds on k are essentially best possible, and we make a first step towards understanding the behavior of the process in the intermediate regime.

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“…There are numerous papers about processes such as these, including Erdős, Suen and Winkler [99], Bollobás and Riordan [46], Osthus and Taraz [156], Ruciński and Wormald [171,172,173] and Greenhill, Ruciński and Wormald [118]. For example, in the second process the probability of deleting a particular edge from H t may be proportional to the number of triangles containing it.…”

Section: Classical Modelsmentioning

confidence: 99%