Avoidance of a giant component in half the edge set of a random graph

Authors:
Tom Bohman;Alan Frieze;Nicholas C. Wormald
Affiliations:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213;Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213;Department of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia
Venue:
Random Structures & Algorithms
Year:
2004

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Abstract

Let e1, e2, … be a sequence of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e., we sample with replacement). Our goal is to choose, for m as large as possible, a subset E ⊆ {e1, e2, …, e2m}, |E| = m, such that the size of the largest component in G = ([n], E) is o(n) (i.e., G does not contain a giant component). Furthermore, the selection process must take place on-line; that is, we must choose to accept or reject on ei based on the previously seen edges e1, …, ei-1.We describe an on-line algorithm that succeeds whp for m = .9668n. A sequence or events En is said to occur with high probability (whp) if limn → ∞ Pr(En) = 1. Furthermore, we find a tight threshold for the off-line version of this question; that is, we find the threshold for the existence of m out of 2m random edges without a giant component. This threshold is m = c*n where c* satisfies a certain transcendental equation, c* ∈ [.9792, .9793]. We also establish new upper bounds for more restricted Achlioptas processes. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 25, 2004Supported in part by NSF Grant DMS-0100400.Supported in part by NSF Grant CCR-9818411.Research supported in part by the Australian Research Council and in part by Carnegie Mellon University Funds.