On the degrees of vertices in A bichromatic random graph

Pałka, Zbigniew

doi:10.1007/bf01850725

Cited by 6 publications

(2 citation statements)

References 5 publications

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“…In the other two probability models on bipartite graphs, G p and G k , two types of results are known: those on the minimum and maximum degrees [5,7,36] and those on the number of vertices with a given degree [23,33,34]. For results in the digraph counterpart G p see [37] (and below).…”

Section: Historical Notesmentioning

confidence: 99%

Degree sequences of random digraphs and bipartite graphs

McKay

Skerman²

2016

Journal of Combinatorics

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We investigate the joint distribution of the vertex degrees in three models of random bipartite graphs. Namely, we can choose each edge with a specified probability, choose a specified number of edges, or specify the vertex degrees in one of the two colour classes.This problem can alternatively be described in terms of the row and sum columns of random binary matrix or the in-degrees and out-degrees of a random digraph, in which case we can optionally forbid loops. It can also be cast as a problem in random hypergraphs, or as a classical occupancy, allocation, or coupon collection problem.In each case, provided the two colour classes are not too different in size nor the number of edges too low, we define a probability space based on independent binomial variables and show that its probability masses asymptotically equal those of the degrees in the graph model almost everywhere. The accuracy is sufficient to asymptotically determine the expectation of any joint function of the degrees whose maximum is at most polynomially greater than its expectation.

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Section: Historical Notesmentioning

confidence: 99%

Degree sequences of random digraphs and bipartite graphs

McKay

Skerman²

2016

Journal of Combinatorics

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“…[l]- [5], [7]- [9]). A similar subject for other models of random graphs has been investigated in [10]- [13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ^ 1 is fixed) and the ith largest degree in a sparse random graph K np , i.e. when p = p(n) = o(l).…”

Section: Introductionmentioning

confidence: 99%

Some remarks about extreme degrees in a random graph

Pałka

1986

Math. Proc. Camb. Phil. Soc.

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Let Kn, p be a random subgraph of a complete graph Kn obtained by removing edges, each with the same probability q = 1 – p, independently of all other edges (i.e. each edge remains in Kn, p with probability p). Very detailed results devoted to probability distributions of the number of vertices of a given degree, as well as of the extreme degrees of Kn, p, have already been obtained by many authors (see e.g. [l]–[5], [7]–[9]). A similar subject for other models of random graphs has been investigated in [10]–[13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ≥ 1 is fixed) and the ith largest degree in a sparse random graph Kn, p, i.e. when p = p(n) = o(1).

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