For any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon \in (0,+\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, consider the subset <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R} \times [0,\varepsilon ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> of the Euclidean plane; it is called a layer or a strip. In 1998, B. Bauslaugh determined the minimal width <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for which the unit distance graph of such a layer contains a cycle of a given odd length k. In this paper, we show that for any two distinct values <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varepsilon _1,\varepsilon _2 \in (0,+\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ε</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>ε</mi> <mn>2</mn> </msub> <mo>∈</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the unit distance graphs of the layers <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R} \times [0,\varepsilon _1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>ε</mi> <mn>1</mn> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \mathbb {R} \times [0,\varepsilon _2]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>ε</mi> <mn>2</mn> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> are non-isomorphic. We also prove a multidimensional analogue of this theorem for the layers <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}^n \times [0,\varepsilon ]^m \subset \mathbb {R}^{n+m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> <mi>m</mi> </msup> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> equipped with the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$l_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-metric. The third main result of this paper is that, for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n \geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\varepsilon > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, any automorphism of the unit distance graph of the layer <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {R}^n \times [0,\varepsilon ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an isometry. This result is related to the Beckman–Quarles theorem (1953), which states that any map of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> preserving unit distances is an isometry, and to its rational analogue recently proved by A. Sokolov (2023).

Isomorphisms of Unit Distance Graphs of Layers

期刊论文
发表日期：

2026年7月27日

查看全文

Artur Bikeev

摘要

For any $\varepsilon \in (0,+\infty )$ $ε \in (0, + \infty)$ , consider the subset $\mathbb {R} \times [0,\varepsilon ]$ $R \times [0, ε]$ of the Euclidean plane; it is called a layer or a strip. In 1998, B. Bauslaugh determined the minimal width $\varepsilon \in (0,1)$ $ε \in (0, 1)$ for which the unit distance graph of such a layer contains a cycle of a given odd length k. In this paper, we show that for any two distinct values $\varepsilon _1,\varepsilon _2 \in (0,+\infty )$ $ε_{1}, ε_{2} \in (0, + \infty)$ , the unit distance graphs of the layers $\mathbb {R} \times [0,\varepsilon _1]$ $R \times [0, ε_{1}]$ and $ \mathbb {R} \times [0,\varepsilon _2]$ $R \times [0, ε_{2}]$ are non-isomorphic. We also prove a multidimensional analogue of this theorem for the layers $\mathbb {R}^n \times [0,\varepsilon ]^m \subset \mathbb {R}^{n+m}$ $R^{n} \times {[0, ε]}^{m} \subset R^{n + m}$ equipped with the $l_p$ $l_{p}$ -metric. The third main result of this paper is that, for $n \geqslant 2$ $n ⩾ 2$ and $\varepsilon > 0$ $ε > 0$ , any automorphism of the unit distance graph of the layer $\mathbb {R}^n \times [0,\varepsilon ]$ $R^{n} \times [0, ε]$ is an isometry. This result is related to the Beckman–Quarles theorem (1953), which states that any map of $\mathbb {R}^n$ $R^{n}$ preserving unit distances is an isometry, and to its rational analogue recently proved by A. Sokolov (2023).