Showing papers by "Van Vu published in 2001"
TL;DR: Results are obtained on many of the properties of a random d-regular graph when d=d(n) grows more quickly than .
Abstract: Random d-regular graphs have been well studied when d is fixed and the number of vertices goes to infinity. We obtain results on many of the properties of a random d-regular graph when d=d(n) grows more quickly than . These properties include connectivity, hamiltonicity, independent set size, chromatic number, choice number, and the size of the second eigenvalue, among others. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 346–363, 2001.
138 citations
TL;DR: A general concentration result is proved which contains a partial answer to each of the questions motivated by the following questions: what is the upper tail probability of YH around its mean?
Abstract: Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions.• What is the upper tail probability Pr(YH ≥ (1 + e)E(YH))?• For which λ does YH have sub-Gaussian behaviour, namely***** Insert formula here *****where c is a positive constant?• Fixing λ = ω(1) in advance, find a reasonably small tail T = T(λ) such that***** Insert formula here *****We prove a general concentration result which contains a partial answer to each of these questions. The heart of the proof is a new martingale inequality, due to J. H. Kim and the present author [13].
95 citations
TL;DR: The main result states that for every k?2 and for all values of the edge probability p=p(n) down to p=O(n?k+1) the ratio between the choice number and the chromatic number of H(k, n, p) does not exceed k1/(k?1) asymptotically.
21 citations
TL;DR: This paper defines and analyze convergence of the geometric random walks, which are certain random walks on vector spaces over finite fields, and shows that the behavior of such walks is given by certain random matroid processes.
17 citations
TL;DR: Several recent results and questions concerning the list chromatic number are presented, focusing on locally sparse graphs and random graphs.