On Independent Spanning Trees in Random and Pseudorandom Graphs

TL;DR

This work resolves the independent spanning trees conjecture in two broad regimes: random graphs $G(n,p)$ for a wide density range and deterministic yet pseudorandom $(n,d,λ)$-graphs under mild spectral assumptions. The authors combine intricate probabilistic constructions with a flexible ‘nice collection of trees’ framework to upgrade local seed structures into globally spanning, vertex-disjoint root-to-vertex paths. In the random regime, they show whp there are $δ(G)$ ISTs rooted at every vertex for $C\log n/n\le p\le 0.99$, aligning with the connectivity threshold. In the pseudorandom setting, they leverage the Friedmann–Pippenger extendability paradigm and the Expander Mixing Lemma to obtain $(1-\varepsilon)d$ ISTs per root when $d/λ=\omega(\log d)$ and $d=o(n/\log^2 n)$, yielding asymptotic results for random $d$-regular graphs as well. Together, the results provide near-optimal, broad-coverage validation of the Zehavi–Itai conjecture and advance the design of reliable, fault-tolerant networks based on ISTs.

Abstract

In 1989, Zehavi and Itai conjectured that every $k$-connected graph contains $k$ independent spanning trees rooted at any prescribed vertex $r$. That is, for each vertex $v$, the unique $r$-$v$ paths within these $k$ spanning trees are internally disjoint. This fundamental problem has received much attention, in part motivated by its applications to network reliability, but despite that has only been resolved for $k \le 4$ and certain restricted graph families. We establish the conjecture for almost all graphs of essentially any relevant density. Specifically, we prove that there exists a constant $C > 1$ such that, with high probability, the random graph $G(n,p)$ contains $δ(G)$ independent spanning trees rooted at any vertex whenever $C \log n/n \leq p < 0.99$. Since the lower bound on $p$ coincides (up to the constant $C$) with the connectivity threshold of $G(n,p)$, this result is essentially optimal. In addition, we show that $(n,d,λ)$-graphs with fairly mild bounds on the spectral ratio $d/λ$ contain $(1-o(1))d$ independent spanning trees rooted at each vertex, thereby settling the conjecture asymptotically for random $d$-regular graphs as well.

Theorems & Definitions (39)

• Conjecture 1.1: Zehavi-Itai, '89
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Chernoff bounds
• Lemma 2.2
• proof
• Lemma 2.3
• proof