Limit theorems for complete subgraphs of random graphs

Schürger, Klaus

doi:10.1007/bf02018372

Cited by 31 publications

(16 citation statements)

References 3 publications

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“…Lower bounds on the number of cliques in a graph have also been obtained [4,13,14,22,23,24,25]. The number of cliques in a random graph has been studied [3,29,37]. Bounds on the number of cliques in a graph have recently been applied in the analysis of an algorithm for finding small separators [32] and in the enumeration of minor-closed families [28].…”

Section: )Nmentioning

confidence: 99%

On the Maximum Number of Cliques in a Graph

Wood

2007

Graphs and Combinatorics

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Abstract.A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree ∆; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K 5 -minor or no K 3,3 -minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2).

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Section: )Nmentioning

confidence: 99%

On the Maximum Number of Cliques in a Graph

Wood

2007

Graphs and Combinatorics

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“…(см. [6]- [9]). В следующем пункте мы определим исследуемую нами модель случайных графов, после чего сформулируем основной результат данной работы.…”

Section: постановка задачи и формулировка основного результатаunclassified

О Вероятности Вхождения Копии Фиксированного Графа В Случайный Дистанционный Граф

Zhukovskii¹

2012

Matematicheskie Zametki

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“…That is, when p(n) has just exceeded the threshold for the existence of G-subgraphs, Xl, converges to a Poisson distribution if and only if G is strictly balanced. Note that the proofs (see, for example, [3,14]) at some point rely on the fact that all G-subgraphs of Kn,p(n) are nonoverlapping. In other words, if we use X , to denote the number of nonoverlapping G-subgraphs of Kn,p(n), then the above theorem still holds with Xl, replaced by X , .…”

Section: Introductionmentioning

confidence: 99%

A correlation inequality and a poisson limit theorem for nonoverlapping balanced subgraphs of a random graph

Suen

1990

Random Struct Algorithms

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We consider non-overlapping subgraphs of fixed order in the random graph K n , p ( n ) .Fix a strictly strongly balanced graph G. A subgraph of Kn,p(n) isomorphic to G is called a G-subgraph. Let X , be the number of G-subgraphs of Kn,p(n) vertex disjoint to all other G-subgraphs. We show that if E[Xn]+m as n + m , then X,IE[X,] converges to 1 in probability. Also, if E[X,]+ c as n+ m, then X , satisfies a Poisson limit theorem. ThePoisson limit theorem is shown using a correlation inequality similar to those appeared in Janson, tuczak, and Rucinski [8] and Boppana and Spencer [4].

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Limit theorems for complete subgraphs of random graphs

Cited by 31 publications

References 3 publications

On the Maximum Number of Cliques in a Graph

On the Maximum Number of Cliques in a Graph

О Вероятности Вхождения Копии Фиксированного Графа В Случайный Дистанционный Граф

A correlation inequality and a poisson limit theorem for nonoverlapping balanced subgraphs of a random graph

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