On the random minimum edge-disjoint spanning trees problem
Dmitry Shabanov, Nikita Zvonkov
TL;DR
This work solves the minimum-weight problem for the union of $k$ edge-disjoint spanning trees in a complete graph with IID edge weights for all $k>2$, extending the classical $k=1,2$ results. It develops a matroid-based, Kruskal-like greedy framework tailored to $k$-deeply connected components and connects the limit weight to a family of functions $eta_k(c)$ derived from Poisson tails, yielding $W_k o rac{1}{2a} abla ext{( ext{integral})}$ of $x(1-eta_k^2(x))$ over $[0, o ty)$. The paper also proves a structural result for sparse random graphs: a giant $k$-deeply connected component emerges in $G(n, c/n)$ for $c>2c'_k$, with the $(k+1)$-core exhibiting a tight rank-density relationship, advancing the understanding of the interplay between core structure and edge-disjoint spanning forests. Together, these results illuminate both the probabilistic structure of random weighted graphs and the algorithmic behavior of optimal unions of $k$ edge-disjoint trees, with implications for network design under randomness.
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.