Phase transition of the minimum degree random multigraph process

Authors:
Mihyun Kang;Taral Guldahl Seierstad
Affiliations:
Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 Berlin, Germany;Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 Berlin, Germany
Venue:
Random Structures & Algorithms
Year:
2007

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Abstract

We study the phase transition of the minimum degree multigraph process. We prove that for a constant hg ≈ 0.8607, with probability tending to 1 as n → ∞, the graph consists of small components on O(log n) vertices when the number of edges of a graph generated so far is smaller than hgn, the largest component has order roughly n2/3 when the number of edges added is exactly hgn, and the graph consists of one giant component on Θ(n) vertices and small components on O(log n) vertices when the number of edges added is larger than hgn. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007