AI 中文总结
针对任意固定图H的无小子图图，本文证明仅需3棵树即可构成具有常数拉伸的树覆盖，该树数量匹配已知下界，结果通过树覆盖与Assouad–Nagata维数的关联及Liu的维数界推导得出。
AI 中文摘要
在本短文中，我们证明对于任意固定图H，H-无小子图图（即不含H作为子图的图）存在由3棵树构成的树覆盖，且该覆盖具有常数拉伸。所用树的数量与Chen、Tan和Xu近期给出的下界一致，他们证明了环面网格实现常数拉伸至少需要3棵树。我们的结果通过建立树覆盖与Assouad–Nagata维数之间的联系得到，随后调用了Liu近期关于无小子图度量的维数界。
英文摘要
In this short note, we show that $H$-minor-free graphs have a tree cover with $3$ trees and constant stretch for any fixed graph $H$. The number of trees matches the recent lower bound by Chen, Tan, and Xu who showed that a toroidal grid requires at least $3$ trees for constant stretch. Our result is obtained by establishing a connection between tree covers and Assouad--Nagata dimension and then invoking the recent dimension bound for minor-free metrics by Liu.
