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Limit theorems for the weights and the degrees in anN-interactions random graph model

100%
Fazekas I., Porvázsnyik B.
Open Mathematics
|
2016
|
tom 14
|
nr 1
414-424
EN
A random graph evolution based on interactions of N vertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of an M-clique means the number of its interactions. The asymptotic behaviour of the weight of a fixed M-clique is studied. Asymptotic theorems for the weight and the degree of a fixed vertex are also presented. Moreover, the limits of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.
2
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Ramsey Properties of Random Graphs and Folkman Numbers

100%
Rödl V., Ruciński A., Schacht M.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
|
nr 3
755-776
EN
For two graphs, G and F, and an integer r ≥ 2 we write G → (F)r if every r-coloring of the edges of G results in a monochromatic copy of F. In 1995, the first two authors established a threshold edge probability for the Ramsey property G(n, p) → (F)r, where G(n, p) is a random graph obtained by including each edge of the complete graph on n vertices, independently, with probability p. The original proof was based on the regularity lemma of Szemerédi and this led to tower-type dependencies between the involved parameters. Here, for r = 2, we provide a self-contained proof of a quantitative version of the Ramsey threshold theorem with only double exponential dependencies between the constants. As a corollary we obtain a double exponential upper bound on the 2-color Folkman numbers. By a different proof technique, a similar result was obtained independently by Conlon and Gowers.
3
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Non-hyperbolicity in random regular graphs and their traffic characteristics

64%
Tucci G.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1593-1597
EN
In this paper we prove that random d-regular graphs with d ≥ 3 have traffic congestion of the order O(n logd−13 n) where n is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically δ-hyperbolic for any non-negative δ almost surely as n → ∞.
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