AI 中文总结
研究乘积图的树宽与其因子树宽的关系，改进了关于树宽乘积的界，还证明了路径宽等相关参数的乘积不等式，并应用结果表明扩张图乘积有大的子图是扩张图。
AI 中文摘要
树宽是衡量图有多‘像树’的标准度量。本文研究乘积图的树宽如何依赖于其因子的树宽。Kozawa、Otachi和Yamazaki[2014]以及Hickingbotham和Wood[2025]独立证明了对于所有图\(G\)和\(H\)，\(\text{tw}(G\boxtimes H)\geq (\text{tw}(G)+1)\text{had}(H)-1\)，其中\(\text{had}(H)\)是\(H\)的哈迪格数。我们将此界改进为\(\text{tw}(G\boxtimes H)\geq (\text{tw}(G)+1)(\text{tw}(H)+1)-1\)，解决了Hickingbotham和Wood的一个开放问题。我们还证明了路径宽、笛卡尔积和严格荆棘数的类似乘积不等式，严格荆棘数是与树宽相关的参数。作为我们结果的一个应用，我们表明扩张图的乘积有大的子图是扩张图。
英文摘要
Treewidth is the standard measure for how ``tree-like'' a graph is. This paper studies how the treewidth of a product graph depends on the treewidth of its factors. Kozawa, Otachi, and Yamazaki [2014] and Hickingbotham and Wood [2025] independently showed that $\text{tw}(G\boxtimes H)\geq (\text{tw}(G)+1)\text{had}(H)-1$ for all graphs $G$ and $H$, where $\text{had}(H)$ is the Hadwiger number of $H$. We improve this bound to $\text{tw}(G\boxtimes H)\geq (\text{tw}(G)+1)(\text{tw}(H)+1)-1$, thereby solving an open problem of Hickingbotham and Wood. We also prove analogous product inequalities for pathwidth, Cartesian products, and strict bramble number, which is a parameter that is tied to treewidth. As an application of our results, we show that products of expanders have large subgraphs that are expanders.
Comments13 pages
