On Independent Spanning Trees in Random and Pseudorandom Graphs
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916980330594304 |
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| author | Draganić, Nemanja Frankston, Keith Krivelevich, Michael Pokrovskiy, Alexey Yepremyan, Liana |
| author_facet | Draganić, Nemanja Frankston, Keith Krivelevich, Michael Pokrovskiy, Alexey Yepremyan, Liana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26401 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Independent Spanning Trees in Random and Pseudorandom Graphs Draganić, Nemanja Frankston, Keith Krivelevich, Michael Pokrovskiy, Alexey Yepremyan, Liana Combinatorics 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. |
| title | On Independent Spanning Trees in Random and Pseudorandom Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.26401 |