On the Length of a Random Minimum Spanning Tree

Abstract

We study the expected value of the lengthLnof the minimum spanning tree of the complete graphKnwhen each edgeeis 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$$ \mathbb{E}(L_n)=\zeta(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}, $$wherec1,c2are explicitly defined constants.