On the random minimum edge-disjoint spanning trees problem

TL;DR

This paper investigates the asymptotic behavior of the minimal weight union of k edge-disjoint spanning trees in complete graphs with random weights, extending known results to all k > 2 and exploring related sparse graph structures.

Contribution

It extends the known limits of minimal weight unions of edge-disjoint spanning trees to any number of trees greater than two in random complete graphs.

Findings

Derived the limit value for the minimal weight union of k edge-disjoint spanning trees for all k>2.

Extended previous results from k=1,2 to all k>2 in random weighted complete graphs.

Proved a related structural result for sparse random graphs.

Abstract

It is well known that finding extremal values and structures can be hard in weighted graphs. However, if the weights are random, this problem can become way easier. In this paper, we examine the minimal weight of a union of $k$ edge-disjoint trees in a complete graph with independent and identically distributed edge weights. The limit of this value (for a given distribution) is known for $k=1,2$. We extend these results and find the limit value for any $k>2$. We also prove a related result regarding the structure of sparse random graphs.