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On the Length of a Random Minimum Spanning Tree

Part of: Graph theory

Published online by Cambridge University Press:  23 January 2015

COLIN COOPER ,
ALAN FRIEZE ,
NATE INCE ,
SVANTE JANSON  and
JOEL SPENCER
COLIN COOPER
Affiliation:
Department of Computer Science, King's College, University of London, London WC2R 2LS, UK (e-mail: colin.cooper@kcl.ac.uk)
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA15217, USA (e-mail: alan@random.math.cmu.edu, incenate@gmail.com)
NATE INCE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA15217, USA (e-mail: alan@random.math.cmu.edu, incenate@gmail.com)
SVANTE JANSON
Affiliation:
Department of Mathematics, Uppsala University, SE-75310 Uppsala, Sweden (e-mail: svante@math.uu.se)
JOEL SPENCER
Affiliation:
Courant Institute, New York, NY 10012, USA (e-mail: spencer@cims.nyu.edu)
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Abstract

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We study the expected value of the length Ln of the minimum spanning tree of the complete graph Kn when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that limn→∞$\mathbb{E}$(Ln) = ζ(3) and show that

$$\mathbb{E}(L_n)=\zeta(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}},$$
where c1, c2 are explicitly defined constants.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C85: Graph algorithms

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 1: Oberwolfach Special Issue Part 2 , January 2016 , pp. 89 - 107
Copyright
Copyright © Cambridge University Press 2015 

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