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Keywords:

  • random graphs;
  • Achlioptas process;
  • differential equation method;
  • phase transition

Abstract

The Erdős-Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erdős and Rényi states that the Erdős-Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erdős and Rényi, various random graph models have been introduced and studied. In this paper we study the Bohman-Frieze process, a simple modification of the Erdős-Rényi process.

The Bohman-Frieze process also begins with an empty graph on n vertices. At each step two random edges are presented, and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman-Frieze process. We show that it has a qualitatively similar phase transition to the Erdős-Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time tc − ϵ (that is, when the number of edges are (tc − ϵ)n/2) are trees or unicyclic components and that the largest component is of size Ω(ϵ-2log n). Further, at tc + ϵ, all components apart from the giant component are trees or unicyclic and the size of the second-largest component is Θ(ϵ-2log n). Each of these results corresponds to an analogous well-known result for the Erdős-Rényi process. Our proof techniques include combinatorial arguments, the differential equation method for random processes, and the singularity analysis of the moment generating function for the susceptibility, which satisfies a quasi-linear partial differential equation. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013