Isomorphisms of unit distance graphs of layers

TL;DR

This work studies isomorphisms of unit-distance graphs on layered spaces $L(n,m,p,\u03b5)$, showing that the unit-distance graph structure uniquely determines the width parameter $\u03b5$ for a broad range of dimensions and norms. It establishes a planar result using $(N,M)$-combs to distinguish strips by width, and extends this rigidity to higher dimensions under smooth $l_p$ norms with $p\in(1,\infty)$. Additionally, it proves a Beckman--Quarles–type rigidity: for $n\ge 2$, every automorphism of the graph of $L(n,1,2,\u03b5)$ is an isometry. Collectively, the results connect layer geometry with distance-preserving mappings and Aleksandrov-type questions in a graph-theoretic setting, highlighting intrinsic width-based invariants in unit-distance graphs.

Abstract

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 $φ$ 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.

Figures (15)

• Figure 1: The rectangle $a'b"cd$
• Figure 2: The shortest path from $a'$ to $b"$ in $G_i$.
• Figure 3: Proof of the implication $(\neg\Omega)\Leftarrow(\neg\Gamma)$. The point $y$ is closer to $z_m$ than $x'$ and $x"$.
• Figure 4: Proof of the implication $(\Omega)\Leftarrow(\Gamma)$. The perpendicular bisectors to the segments $[x',y]$ and $[x",y]$ separate $z_m$ from $y$ for sufficiently large $m$.
• Figure 5: a). The case $y \in [x,z]$, $||a-x||_2 \geqslant ||a-y||_2$. b) The case $f(x) \in [f(y), f(z)]$, $\max(\rho_{1}(b,f(x)), \rho_{1}(b,f(z))) < \rho_{1}(b,f(y))$

Theorems & Definitions (49)

• Theorem 1
• Theorem 2
• Remark
• Theorem 3
• Proposition 5
• proof
• Corollary 6
• Lemma 7
• proof
• Lemma 8