Spanning trees in sparse expanders Article Swipe
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· 2022
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.2211.04758
· OA: W4308758091
Given integers $n\ge Δ\ge 2$, let $\mathcal{T}(n, Δ)$ be the collection of all $n$-vertex trees with maximum degree at most $Δ$. A question of Alon, Krivelevich and Sudakov in 2007 asks for determining the best possible spectral gap condition forcing an $(n, d,λ)$-graph to be $\mathcal{T}(n, Δ)$-universal, namely, it contains all members of $\mathcal{T}(n, Δ)$ as a subgraph simultaneously. In this paper we show that for sufficiently large integer $n$ and all $Δ\in \mathbb{N}$, every $(n, d,λ)$-graph with \[ λ\le\frac{d}{2Δ^{5\sqrt{\log n}}} \] is $\mathcal{T}(n, Δ)$-universal. As an immediate corollary, this implies that Alon's ingenious construction of triangle-free sparse expander is $\mathcal{T}(n, Δ)$-universal, which provides an explicit construction of such graphs and thus solves a question of Johannsen, Krivelevich and Samotij. Our main result is formulated under a much more general context, namely, the $(n,d)$-expanders. More precisely, we show that there exist absolute constants $C,c>0$ such that the following statement holds for sufficiently large integer $n$. (1).For all $Δ\in \mathbb{N}$, every $(n, Δ^{5\sqrt{\log n}})$-expander is $\mathcal{T}(n, Δ)$-universal. (2).For all $Δ\in \mathbb{N}$ with $Δ\le c\sqrt{n}$, every $(n, CΔn^{1/2})$-expander is $\mathcal{T}(n, Δ)$-universal. Both results significantly improve a result of Johannsen, Krivelevich and Samotij, and have further implications in locally sparse expanders and Maker-Breaker games that also improve previously known results drastically.
