On the length of a random minimum spanning tree

TL;DR

This paper refines the understanding of the expected length of a minimum spanning tree in a complete graph with random edge weights, providing precise asymptotic expansions beyond the known limit.

Contribution

It improves previous results by deriving detailed asymptotic formulas for the expected MST length, including explicit constants and correction terms.

Findings

Expected MST length converges to ζ(3) as n→∞

Derived explicit formulas for the first correction terms in the asymptotic expansion

Provided precise constants c_1 and c_2 for the asymptotic formula

Abstract

We study the expected value of the length $L_n$ of the minimum spanning tree of the complete graph $K_n$ when each edge $e$ is given an independent uniform $[0,1]$ edge weight. We sharpen the result of Frieze \cite{F1} that $\lim_{n\to\infty}\E(L_n)=\z(3)$ and show that $\E(L_n)=\z(3)+\frac{c_1}{n}+\frac{c_2+o(1)}{n^{4/3}}$ where $c_1,c_2$ are explicitly defined constants.