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The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdo&huml;s over the past half century. Here I will explore a particular set of problems - all dealing(More)
Single-nucleotide polymorphisms (SNPs) are the most frequent type of variation in the human genome, and they provide powerful tools for a variety of medical genetic studies. In a large-scale survey for SNPs, 2.3 megabases of human genomic DNA was examined by a combination of gel-based sequencing and high-density variation-detection DNA chips. A total of(More)
The k-core of a graph is the largest subgraph with minimum degree at least k. For the Erd˝ os-Rényi random graph G(n, m) on n vertices, with m edges, it is known that a giant 2-core grows simultaneously with a giant component, that is when m is close to n/2. We show that for k ≥ 3 , with high probability, a giant k-core appears suddenly when m reaches c k(More)
Recently, Barabási and Albert [2] suggested modeling complex real-world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier ver-tices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabási(More)
Given a graph G and a subset S of the vertex set of G, the discrepancy of S is defined as the difference between the actual and expected numbers of the edges in the subgraph induced on S. We show that for every graph with n vertices and e edges, n < e < n(n-1)/4, there is an n/2-element subset with the discrepancy of the order of magnitude of V ne. For(More)
Networks in which the formation of connections is governed by a random process often undergo a percolation transition, wherein around a critical point, the addition of a small number of connections causes a sizable fraction of the network to suddenly become linked together. Typically such transitions are continuous, so that the percentage of the network(More)