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On the evolution of random graphs. (English) Zbl 0103.16301
A random graph Γ n,N is a undirected finite graph without parallel edges and slings. Γ n,N has n points P 1 ,···,P n and N edges (P i ,P j ), which are chosen at random so that all n 2 N=C n,N possible choices are supposed to be equiprobable. Let be P n,N (A)=A n,N /C n,N the probability that Γ n,N has the property A, where A n,N denotes the number of graphs with the given points P 1 ,···,P n , with N edges (P i ,P j ) and with the property A. Γ n,N is studied under the condition that N is increased, i.e. if N is equal, or asymptotically equal, to a given function N(n) of n. For many properties A there is shown that there exists a ”threshold function” A(n) of the property A tendig monotonically to + for n+ such that lim n+ P n,N(n) (A)=0 or =1 if lim n+ N(n) A(n)=0 or =+. A(n) is a ”regular threshold function” of A if there exists a probability distribution function F(x) such that lim n+ P n,N(n) (A)=F(x) if lim n+ N(n) A(n)=x, where 0<x<+ and x is a point of continuity of F(x). The investigated properties are as follows: the presence of certain subgraphs (e. g. trees, complete subgraphs, cycles, etc.) or connectedness, number of components etc. The results are of the following type: Theorem 3a. Suppose that N(n)cn, where c>0. Let γ k denote the number of cycles of order k contained in Γ n,N (k=3,4,···). Then we have lim n+ P n,N(n) (γ k =j)=λ j e -λ /j!, where j=0,1,··· and λ=(2c) k /2k. Thus the threshold distribution corresponding to the threshold function A(n)=n for the property that the graph contains a cycle of order k is 1-e -(2c) k /2k .
Reviewer: K.Čulik

05C80Random graphs