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We study the k-core of a random (multi)graph on n ver-tices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k-core is empty, and other conditions that imply that with high probability the… (More)

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high… (More)

Talagrand (Publ. Math. Inst. Hautes Etudes Sci. 81 (1995) 73) gave a concentration inequality concerning permutations picked uniformly at random from a symmetric group, and this was extended in McDiarmid (Combin. Probab. Comput. 11 (2002) 163) to handle permutations picked uniformly at random from a direct product of symmetric groups. Here we extend these… (More)

We define the decision problem data arrangement, which involves arranging the vertices of a graph G at the leaves of a d-ary tree so that a weighted sum of the distances between pairs of vertices measured with respect to the tree topology is at most a given value. We show that data arrangement is strongly NP-complete for any fixed d ≥ 2 and explain the… (More)

We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter… (More)

Consider partitions of the vertex set of a graph G into two sets with sizes differing by at most 1: the bisection width of G is the minimum over all such partitions of the number of ''cross edges'' between the parts. We are interested in sparse random graphs Ž. G with edge probability crn. We show that, if c) ln 4, then the bisection width is ⍀ n n, c r n… (More)

We compare the performance of a variant of the standard Dynamic Alternative Routing (DAR) technique commonly used in telephone and ATM networks to a path selection algorithm that is based on the balanced allocations principle [4, 18]-the Balanced Dynamic Alternative Routing (BDAR) algorithm. While the standard technique checks alternative routes… (More)

The 2-dimensional Hamming graph H(2, n) consists of the n 2 vertices (i, j), 1 ≤ i, j ≤ n, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2, n) in percolation with edge probability p, so that the average degree 2(n − 1)p = 1 +. Previous work [5] had shown that in the barely supercritical region n −2/3 ln… (More)

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