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The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d

Part of: Graph theory Combinatorial probability Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Martin J. B. Appel  and
Ralph P. Russo
Martin J. B. Appel*
Affiliation:
United Technologies Research Center
Ralph P. Russo*
Affiliation:
University of Iowa
*
∗ Postal address: United Technologies Research Center, East Hartford, CT 06108, USA.
∗∗ Postal address: Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, USA.
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Abstract

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.

Keywords

RANDOM GRAPHMAXIMUM VERTEX DEGREE

MSC classification

Primary: 05C80: Random graphs
Secondary: 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 3 , September 1997 , pp. 567 - 581
Copyright
Copyright © Applied Probability Trust 1997 

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