标题： 单位距离图层的同构 显示英文标题

标题： Isomorphisms of unit distance graphs of layers

摘要： 对于任意的$\varepsilon \in (0,+\infty)$，考虑欧几里得平面中的度量空间$\mathbb{R} \times [0,\varepsilon]$，称为层或条带。 B. Baslaugh 在1998年找到了一个条带的最小宽度$\varepsilon \in (0,1)$，使得其单位距离图包含给定奇数长度$k$的循环。 本文的主要结果之一是两个层的单位距离图$\mathbb{R} \times [0,\varepsilon_1], \mathbb{R} \times [0,\varepsilon_2]$对于任何不同的值$\varepsilon_1,\varepsilon_2 \in (0,+\infty)$都是非同构的事实。 我们还得到了该定理的一个多维类比。 对于给定的$n,m \in \mathbb{N}, p \in (1,+\infty), \varepsilon \in (0,+\infty)$，我们说定义在$\mathbb{R}^n \times [0,\varepsilon]^m$上且由$\mathbb{R}^{n+m}$中的$l_p$-范数生成的度量空间距离构成的度量空间是一个层$L(n,m,p,\varepsilon)$。 我们证明了当$\varepsilon_1 \neq \varepsilon_2$时，层$L(n,m,p,\varepsilon_1), L(n,m,p,\varepsilon_2)$的单位距离图是非同构的。 本文的第三个主要结果是定理：对于$n \geq 2, \varepsilon > 0$，层$L = L(n,1,2,\varepsilon) = \mathbb{R}^n \times [0,\varepsilon]$的单位距离图的任意自同构$\phi$都是保距映射。 这与1953年Beckman和Quarles提出的定理有关，该定理指出$\mathbb{R}^n$的任意保单位的映射都是保距映射，并且与A. Sokolov在2023年得出的该定理的有理数版本相关。 显示英文摘要

摘要： For any $\varepsilon \in (0,+\infty)$, consider the metric spaces $\mathbb{R} \times [0,\varepsilon]$ in the Euclidean plane named layers or strips. B. Baslaugh in 1998 found the minimal width $\varepsilon \in (0,1)$ of a layer such that its unit distance graph contains a cycle of a given odd length $k$. The first of the main results of this paper is the fact that the unit distance graphs of two layers $\mathbb{R} \times [0,\varepsilon_1], \mathbb{R} \times [0,\varepsilon_2]$ are non-isomorphic for any different values $\varepsilon_1,\varepsilon_2 \in (0,+\infty)$. We also get a multidimensional analogue of this theorem. For given $n,m \in \mathbb{N}, p \in (1,+\infty), \varepsilon \in (0,+\infty)$, we say that the metric space on $\mathbb{R}^n \times [0,\varepsilon]^m$ with the metric space distance generated by $l_p$-norm in $\mathbb{R}^{n+m}$ is a layer $L(n,m,p,\varepsilon)$. We show that the unit distance graphs of layers $L(n,m,p,\varepsilon_1), L(n,m,p,\varepsilon_2)$ are non-isomorphic for $\varepsilon_1 \neq \varepsilon_2$. The third main result of this paper is the theorem that, for $n \geq 2, \varepsilon > 0$, any automorphism $\phi$ of the unit distance graph of layer $L = L(n,1,2,\varepsilon) = \mathbb{R}^n \times [0,\varepsilon]$ is an isometry. This is related to the Beckman-Quarles theorem of 1953, which states that any unit-preserving mapping of $\mathbb{R}^n$ is an isometry, and to the rational analogue of this theorem obtained by A. Sokolov in 2023.