The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d
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On independent random points U1,· ··,Un distributed uniformly on [0, 1]d, a random graph Gn(x) is constructed in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be empty of edges, a.s.
1997 ◽
Vol 29
(03)
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pp. 567-581
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2011 ◽
Vol 20
(5)
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pp. 763-775
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1982 ◽
Vol 91
(1)
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pp. 119-130
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