环面和克莱因瓶上的4-正则四边形镶嵌图的$l_1$-嵌入
$l_{1}$-embeddability of 4-regular quadrilateral maps on the torus and the Klein bottle
王广富 1丛文超 1徐春晓1
作者信息
- 1. 烟台大学数学与信息科学学院, 烟台, 264005
- 折叠
摘要
\justifying一个连通图$G$若能等距离嵌入到$l_1$-空间中, 则称$G$是$l_1$-嵌入的。 图$Q_{k,m,r}$和$Q_{k,m,e}$是定义在环面上的两类4-正则四边形镶嵌图,它们由路$P_{m}$与路$P_{k}$($k,m\ge1$)的卡氏积构造得到的；而图$Q_{k,m,a}$,$Q_{k,m,b}$和$Q_{k,m,h}$是定义在克莱因瓶上的三类4-正则四边形镶嵌图,它们由路$P_{m}$和圈$C_{k}$ ($k \ge 1, m \ge2$)的卡氏积构造得到的。在本文中,我们证明了除了$\mathcal{Q}_{k,m,0}$, $\mathcal{Q}_{k,2,1}$, $\mathcal{Q}_{k,1,e}$, $Q_{5,2,e}$, $\mathcal{Q}_{2i,2,e}$ ($i \in Z^{+}$), $Q_{2,2,a}$, $Q_{4,2,a}$, $\mathcal{Q}_{2,m,b}$和$Q_{1,2,h}$,其余所有的$Q_{k,m,r}$, $Q_{k,m,e}$, $Q_{k,m,a}$, $Q_{k,m,b}$和$Q_{k,m,h}$ 都不是$l_1$-可嵌入的。
Abstract
\justifyingA connected graph $G$ is said to be $l_1$-embeddable if it can be isometrically embedded into the $l_1$-space. The graphs $Q_{k,m,r}$ and $Q_{k,m,e}$ are two types of 4-regular quadrilateral maps on the torus obtained from Cartesian product path $P_{m}$ and path $P_{k}$($k, m \ge 1$), and the graphs $Q_{k,m,a}$, $Q_{k,m,b}$, $Q_{k,m,h}$ are three types of 4-regular quadrilateral maps on the Klein bottle obtained from Cartesian product path $P_{m}$ and cycle $C_{k}$ ($k \ge 1, m \ge 2$). In this paper, we determine that all the $Q_{k,m,r}$, $Q_{k,m,e}$, $Q_{k,m,a}$, $Q_{k,m,b}$ and $Q_{k,m,h}$ are not $l_1$-embeddable except for $\mathcal{Q}_{k,m,0}$, $\mathcal{Q}_{k,2,1}$, $\mathcal{Q}_{k,1,e}$, $Q_{5,2,e}$, $\mathcal{Q}_{2i,2,e}$ ($i \in Z^{+}$), $Q_{2,2,a}, Q_{4,2,a},\mathcal{Q}_{2,m,b}$ and $Q_{1,2,h}$.关键词
$l_1$-嵌入/4-正则四边形镶嵌图/ 环面/ 克莱因瓶Key words
$l_{1}$-embeddable/ 4-regular quadrilateral maps/ torus/ Klein bottle引用本文复制引用
王广富,丛文超,徐春晓.环面和克莱因瓶上的4-正则四边形镶嵌图的$l_1$-嵌入[EB/OL].(2026-03-24)[2026-03-27].http://www.paper.edu.cn/releasepaper/content/202603-236.学科分类
数学
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