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On the strength of connectedness of a random graph. (English) Zbl 0103.16302

Using the notation of the paper reviewed above the following theorem is proved: If \(N(n) ={1 \over 2} n \log n + {1 \over 2} r n \log \log n + \alpha n + o(n)\), where \(\alpha\) is a real constant and \(r\) a non-negative integer, then \[ \lim_{n \to +\infty} \text{Pr}(c_i(\Gamma_{n,N(n)}) = r) = 1-\exp(-e^{-2\alpha}/r!), \] where \(i=1,2,3\) and \(c_1(G)\) denotes the minimal number of all edges starting from a single point in a given graph \(G\), \(c_2(G)\) or \(c_3 (G)\) denotes the least number \(k\) such that by deleting \(k\) appropriately chosen points or edges the resulting graph is disconected (if \(G\) is complete with \(n\) points one puts \(c_2(G) = n-1\)).
Reviewer: K.Čulik

MSC:

05C40 Connectivity
05C80 Random graphs (graph-theoretic aspects)

Keywords:

topology
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References:

[1] D. König,Theorie der endlichen und unendlichen Graphen (Leipzig, 1936). · JFM 62.0654.05
[2] C. Berge,Théorie des graphes et ses applications (Paris, 1958).
[3] L. R. Ford andD. R. Fulkerson, Maximal flow through a network,Canadian Journal of Math.,8 (1956), p. 399. · Zbl 0073.40203
[4] P. Erdos andA. Rényi, On random graphs. I,Publ. Math. Debrecen,6 (1959), pp. 290–297. · Zbl 0092.15705
[5] P. Erdos andA. Rényi, On the evolution of random graphs,Publ. Math. Inst. Hung. Acad. Sci.,5 (1960), pp. 17–61. · Zbl 0103.16301
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