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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)
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)
18 until the number of vertices left is O(n=d), and color the remaining vertices in an arbitrary manner. 7. The existence of an approximation algorithm based on the spectral method for coloring arbitrary graphs is a question that deserves further investigation (which we do not address here.) Recently, improved approximation algorithms for graph coloring(More)
The standard Erd˝ os-Renyi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n 2 rounds as one time unit, a phase transition occurs at t = 1 when a giant component (one of size constant time n) first appears. Under the influence of statistical mechanics, the investigation of related phase(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)