The strong thin tree conjecture states that every $k$-edge-connected graph $G$ contains an $O(1/k)$-thin spanning tree, meaning a spanning tree which contains at most an $O(1/k)$ fraction of the edges across each cut in $G$. This conjecture is still open despite significant effort; the best current result by Anari and Oveis Gharan shows the existence of an $O(\text{polyloglog}(n)/k)$-thin tree. In this work, we demonstrate that the conjecture is true if one only requires thinness for the set of $η$-near minimum cuts of the graph for $η= 1/40$, in other words, for the set of cuts with fewer than $(1+1/40)k$ edges. Our approach constructs such a tree in polynomial time. To show this, we utilize the structure of near minimum cuts, and in particular the polygon representation of Benczúr and Goemans, to reduce to the previously solved problem of finding a spanning tree that is $O(1/k)$-thin for all sets in a laminar family.

翻译：强薄树猜想指出，每个 $k$ 边连通图 $G$ 都存在一棵 $O(1/k)$ 薄的生成树，即该生成树在 $G$ 的每个割集中至多包含 $O(1/k)$ 比例的边。尽管已有显著研究进展，该猜想仍未解决；目前 Anari 和 Oveis Gharan 得到的最佳结果表明存在一棵 $O(\text{polyloglog}(n)/k)$ 薄的树。本研究证明，若仅需对图中 $η= 1/40$ 的 $η$ 近最小割族（即边数少于 $(1+1/40)k$ 的割集）满足薄性条件，则该猜想成立。本文方法可在多项式时间内构造出此类树。为证明该结论，我们利用近最小割的结构性质，特别是 Benczúr 和 Goemans 提出的多边形表示方法，将问题归结为在层状族中寻找对所有集合 $O(1/k)$ 薄的生成树这一已解决的问题。