Almost universal graphs

  • Authors:

  • Alan Frieze;Michael Krivelevich

  • Affiliations:

  • Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA;Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2006

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Abstract

We study the question as to when a random graph with n vertices and m edges contains a copy of almost all graphs with n vertices and cn/2 edges, c constant. We identify a “phase transition” at c = 1. For c m must grow slightly faster than n, and we prove that m = O(n log log n/log log log n) is sufficient. When c 1, m must grow at a rate m = n1+a, where a = a(c) 0 for every c 1, and a(c) is between $1-{2 \over (1+o(1))c}$ and $1-{1 \over c}$ for large enough c. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006