Journal ArticleDOI
The size ramsey number
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TLDR
In this article, the class of all graphs G which satisfy the Ramsey number G→(G>>\s 1, G>>\s 2) is defined, and the asymptotic behavior of the Ramsey numbers is investigated.Abstract:
Let denote the class of all graphsG which satisfyG→(G
1,G
2). As a way of measuring minimality for members of
, we define thesize Ramsey number ř(G
1,G
2) by
. We then investigate various questions concerned with the asymptotic behaviour ofř.read more
Citations
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Journal ArticleDOI
Explicit construction of linear sized tolerant networks
Noga Alon,Fan Chung +1 more
TL;DR: This paper constructs explicitly graphs with O(m/@e) vertices and maximum degree O(1/@ e^2), such that after removing any (1-@e%) portion of their vertices or edges, the remaining graph still contains a path of length m.
Proceedings ArticleDOI
Sparse partitions
Baruch Awerbuch,David Peleg +1 more
TL;DR: A collection of clustering and decomposition techniques that make possible the construction of sparse and locality-preserving representations for arbitrary networks is presented and several other graph-theoretic structures that are strongly related to covers are discussed.
Book ChapterDOI
On size Ramsey number of paths, trees, and circuits. I
TL;DR: It is demonstrated that random graphs satisfy some interesting Ramsey type properties and are shown to be finite, simple and undirected graphs.
On graphs of ramsey type
Paul Erdös,László Lovász +1 more
TL;DR: In this paper, the authors consider the problem of characterizing those graphs for which F 3 (G,H) for a given G and H can be written, and they show that the problem is in general extremely difficult.
Journal ArticleDOI
The Induced Size-Ramsey Number of Cycles
TL;DR: A result is proved that implies that the induced r -size-Ramsey number of the cycle C l is at most c r l for some constant c r that depends only upon r.
References
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Book
Probabilistic Methods in Combinatorics
TL;DR: In this paper, Erdős [8] showed that the probabilistic method must exist for a graph G(n,.5) to be a random graph, and that a graph satisfying ∧( √ B_s) ≠ ∅ must exist.