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Keywords:

  • universal graphs;
  • random graphs;
  • Erdős-Rényi

Abstract

For any integer n, let equation image be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is equation image-almost-universal if Γ satisifies the following: If G is chosen according to the probability distribution equation image, then G is isomorphic to a subgraph of Γ with probability 1 - equation image. For any p ∈ [0,1], let equation image(n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,vV (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a equation image-almost-universal-graph on n vertices and with O(nmath image)polylog(n) edges, where q = ⌈equation image⌉ for such k ≤ 6, and where q = ⌈equation image⌉ for k ≥ 7. The number of edges is close to the lower bound of Ω(equation image) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010