TY - JOUR

T1 - Almost universal graphs

AU - Frieze, Alan

AU - Krivelevich, Michael

PY - 2006/7

Y1 - 2006/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33745697574&partnerID=8YFLogxK

U2 - 10.1002/rsa.20121

DO - 10.1002/rsa.20121

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AN - SCOPUS:33745697574

VL - 28

SP - 499

EP - 510

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 4

ER -