Almost universal graphs

Alan Frieze, Michael Krivelevich

Research output: Contribution to journalArticlepeer-review


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 < 1, 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 -2/1+o(1))c and 1 -1/c for large enough c.

Original languageEnglish
Pages (from-to)499-510
Number of pages12
JournalRandom Structures and Algorithms
Issue number4
StatePublished - Jul 2006


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