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On the Length of a Random Minimum Spanning Tree

  • COLIN COOPER (a1), ALAN FRIEZE (a2), NATE INCE (a2), SVANTE JANSON (a3) and JOEL SPENCER (a4)...

Abstract

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.

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