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On random graphs i
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On random graphs i   (Citations: 1181)
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    • ...amenable to standard rst and second moment proof techniques along the lines of the textbook proof of Theorem 1 (see e.g. [5] or [9])...
    • ...Part ii) and Theorem 8 then follow analogously to the textbook rst and second moment proof of Theorem 1 (see e.g. [5] or [9])...

    Torsten Mütze, et al. Small subgraphs in random graphs and the power of multiple choices

    • ...n-element vertex set V (see [4]), random k-uniform hypergraphs (see [1]), where Γ= V k , and random subsets of integers, where V ={1, 2, ..., n} (see [2], [8])...
    • ...Kasteleyn and Ginibre (lower bound) and Janson’s inequality (upper bound), see [4], Section 2.2...
    • ...Often, these two bounds asymptotically match under some restrictions on the dependencies among the summands IS. This is, in particular, the case of subgraph counts in random graphs, see [4], Section 3.1...
    • ...Some ad hoc results can be found in [4], [9], [5 ]a nd [6], among others...
    • ...By the same argument as for the unrooted case in [4, Section 3.1], it is easy to show that p=n −1/mR(G) is the threshold for the appearance of an R-rooted copy of G in...

    Svante Janson, et al. Upper tails for counting objects in randomly induced subhypergraphs an...

    • ...All nodes in this cell and primary secure links between these nodes form a subgraph, which can be modeled as an Erdös–Rényi random graph [33], [34]...
    • ...From the above proof, we know that the number of nodes in each cell is and that the average node degree in this subgraph is , which is larger than the logarithm of the number of nodes in the cell, given that . Therefore, by the properties of the Erdös–Rényi random graph [33], [34], this subgraph is connected, i.e., there exists a secure path connecting arbitrary node pairs in the cell...

    Chi Zhang, et al. On the Price of Security in Large-Scale Wireless Ad Hoc Networks

    • ...where the sum runs over all ordered pairs x y, including those with x = y. By Janson’s inequality [25, Theorem 2.18],...

    Béla Bollobás, et al. On covering by translates of a set

    • ...If for any graph S such that H ⊂ S ⊆ G, we have the inequality f (S, H) > 0, then the pair (G, H) is called α�reliable (see [1, 3, 12])...
    • ...If for any graph S such that H ⊆ S ⊂ G, it holds that f (G, S) < 0, then the pair (G, H) is called α�rigid (see [1, 12])...
    • ...Now we formulate a theorem (see [1, 11, 12]) on the num� ber of copies of a strictly balanced graph...

    M. E. Zhukovskii. Zero-one laws for first-order formulas with a bounded quantifier depth

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