Journal ArticleDOI
Cliques in random graphs
Béla Bollobás,Paul Erdös +1 more
- Vol. 80, Iss: 3, pp 419-427
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TLDR
In this paper, the maximal size of a clique and the number of Kr's in a complete graph with n points and m edges is investigated. But the maximal clique is not a maximal complete subgraph.Abstract:
Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined byare independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variablesthe number of Kr's in Gn, andthe maximal size of a clique in Gn.read more
Citations
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Book ChapterDOI
The maximum clique problem
TL;DR: A survey of results concerning algorithms, complexity, and applications of the maximum clique problem is presented and enumerative and exact algorithms, heuristics, and a variety of other proposed methods are discussed.
BookDOI
Complexity and Approximation
Giorgio Ausiello,Alberto Marchetti-Spaccamela,Pierluigi Crescenzi,Giorgio Gambosi,Marco Protasi,Viggo Kann +5 more
Journal ArticleDOI
Networks beyond pairwise interactions: Structure and dynamics
Federico Battiston,Giulia Cencetti,Iacopo Iacopini,Iacopo Iacopini,Vito Latora,Maxime Lucas,Alice Patania,Jean-Gabriel Young,Giovanni Petri +8 more
TL;DR: A complete overview of the emerging field of networks beyond pairwise interactions, and focuses on novel emergent phenomena characterizing landmark dynamical processes, such as diffusion, spreading, synchronization and games, when extended beyond Pairwise interactions.
MonographDOI
Introduction to random graphs
Alan Frieze,Michał Karoński +1 more
TL;DR: All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
Journal ArticleDOI
The chromatic number of random graphs
TL;DR: For a fixed probability p, 0 < p < 1, almost every random graph Gn,p has chromatic number σ 2 + o(1) + σ σ 1/(1 - p)) σ n{log n} as mentioned in this paper.
References
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Journal ArticleDOI
On colouring random graphs
TL;DR: In this paper, it was shown that the number of vertices in the largest complete subgraph of ωn is, with probability one, the same as in this paper.
Journal ArticleDOI
On complete subgraphs of different orders
TL;DR: In this paper, it was shown that the edges of a graph with n ≥ 1 vertices can be covered with at most [n 2/4] edge disjoint triangles and edges if every edge of G is in at least one Gi.