Open Access
Chromatic number of random graphs
Béla Bollobás
- 01 Jan 1988
322
TL;DR: It is shown that for a fixed probabilityp, 0<p<1, almost every random graphGn,p has chromatic number log (1/(1 - p) + o(1) + 1 + o (1)log n, where n is the number of random graphs.
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Abstract: One of the best known and most studied unsol;eed problems in the theory of random graphs is the determination of the chromatic number of almost every random graph. As usual, we write Gp for a random graph with vertex set V= In] = {1, 2 . . . . . n} in which the edges are chosen independently and with probability p=p(n), 0 < p < 1. In most of what follows, we shall take p to be fixed and write d = 1/(1-p). Putting it rather crudely, Grimmett and McDiarmid [8] proved that a. e. Gp is such that its chromatic number X(Gp) satisfies
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Asymptotic theory of finite dimensional normed spaces
Vitali Milman,Gideon Schechtman +1 more
- 01 Jan 1986
TL;DR: The concentration of measure phenomenons in the theory of Normed spaces was discussed in this paper, where the Rademacher projection was applied to the case of finite Dimensional Normed Spaces.
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On Tail Probabilities for Martingales
TL;DR: In this paper, the Laplace transform of the crossing time of a martingale with uniformly bounded increments is shown to have the same distribution as the distribution of crossing times of Brownian motion, even in the tail.
On colouring random graphs
Geoffrey Grimmett,Colin McDiarmid +1 more
- 01 Mar 1975
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.
406
Cliques in random graphs
Béla Bollobás,Paul Erdös +1 more
- 01 Nov 1976
TL;DR: 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.
398