On edge disjoint spanning trees in a randomly weighted complete graph

TL;DR

This paper investigates the expected minimum total weight of multiple edge-disjoint spanning trees in a complete graph with randomly assigned uniform edge weights, providing asymptotic results and explicit constants.

Contribution

It introduces new asymptotic estimates for the minimum total weight of multiple edge-disjoint spanning trees in random weighted complete graphs, especially for the case of two trees.

Findings

Expected minimum weight of k trees approximates k^2 for large k.

Expected minimum weight of two trees tends to a specific constant approximately 4.1704.

Provides explicit asymptotic behavior for the case k=2.

Abstract

Assume that the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ edges weights. We consider the expected minimum total weight $\mu_k$ of $k\geq 2$ edge disjoint spanning trees. When $k$ is large we show that $\mu_k\approx k^2$. Most of the paper is concerned with the case $k=2$. We show that $\m_2$ tends to an explicitly defined constant and that $\mu_2\approx 4.1704288\ldots$.