发表机构
School of Mathematics and Information Sciences, Yantai University; ECOPRO, Institute for Basic Science（烟台大学数学与信息科学学院; 韩国基础科学研究院ECOPRO）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文对环面和克莱因瓶上的正则四边形映射给出了 $\ell_1$-可嵌入性的完整参数级分类，精确刻画了五个族中可嵌入的具体参数，并证明其余均不可嵌入。
AI 中文摘要
一个连通图被称为 $\ell_1$-可嵌入的，如果其最短路径度量允许等距嵌入到 $\ell_1$ 空间中；等价地，在有限情形下，它允许尺度嵌入到超立方体中。本文给出了环面和克莱因瓶上正则四边形映射的 $\ell_1$-可嵌入性的完整参数级分类。我们处理两个环面族 $Q_{k,m,r}$ 和 $Q_{k,m,e}$，它们由有限正方形网格的适当识别得到，并利用已知的克莱因瓶四边形映射之间的同构，处理三个本质的克莱因瓶族 $Q_{k,m,a}$、$Q_{k,m,b}$ 和 $Q_{k,m,h}$。$\ell_1$-可嵌入的环面映射恰好是 $Q_{k,m,0}, Q_{k,2,1}, Q_{k,1,e}, Q_{5,2,e}, Q_{2i,2,e} (i\ge 2)$，而 $\ell_1$-可嵌入的克莱因瓶映射恰好是 $Q_{2,2,a}, Q_{4,2,a}, Q_{2,m,b}, Q_{1,2,h}$。这五个族中的所有其他成员都不是 $\ell_1$-可嵌入的。
英文摘要
A connected graph is called $\ell_1$-embeddable if its shortest-path metric admits an isometric embedding into the $\ell_1$-space; equivalently, in the finite case, it admits a scale embedding into a hypercube. This paper gives a complete parameter-level classification of $\ell_1$-embeddability for regular quadrilateral maps on the torus and the Klein bottle. We treat the two toroidal families $Q_{k,m,r}$ and $Q_{k,m,e}$, obtained from suitable identifications of finite square grids, and, using the known isomorphisms among Klein-bottle quadrilateral maps, the three essential Klein-bottle families $Q_{k,m,a}$, $Q_{k,m,b}$, and $Q_{k,m,h}$. The $\ell_1$-embeddable toroidal maps are exactly $Q_{k,m,0}, Q_{k,2,1}, Q_{k,1,e}, Q_{5,2,e}, Q_{2i,2,e} (i\ge 2)$, and the $\ell_1$-embeddable Klein-bottle maps are exactly $ Q_{2,2,a}, Q_{4,2,a}, Q_{2,m,b}, Q_{1,2,h}.$ All other members of these five families are not $\ell_1$-embeddable.
Comments17 pages
