Bikeev 2025 isomorphisms unit distance graphs layers

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theorem_1: Bikeev's theorem that for widths epsilon_1, epsilon_2 in (0, infinity) the unit distance graphs of the Euclidean strips R x [0, epsilon_1] and R x [0, epsilon_2] are isomorphic if and only if epsilon_1 = epsilon_2.

theorem_2: Bikeev's theorem that for n >= 2, m >= 1, p in (1, infinity) and widths epsilon_1, epsilon_2 in (0, infinity) the unit distance graphs of the layers L(n,m,p,epsilon_1) and L(n,m,p,epsilon_2) are isomorphic if and only if epsilon_1 = epsilon_2.

theorem_3: Bikeev's theorem that for n >= 2 and epsilon > 0 every automorphism of the unit distance graph of the Euclidean layer L(n,1,2,epsilon) = R^n x [0, epsilon] is an isometry, a Beckman-Quarles-type statement.

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Arthur Bikeev, Isomorphisms of unit distance graphs of layers. arXiv preprint (2025). arXiv:2505.07799. The copy read for this card is arXiv:2505.07799v3, dated 23 May 2025.

Bikeev studies unit distance graphs of layers L(n,m,p,epsilon), the metric spaces R^n x [0,epsilon]^m with the distance induced by the l_p-norm on R^(n+m); the case L(1,1,2,epsilon) is the Euclidean strip R x [0,epsilon]. Theorem 1 proves that the unit distance graphs of two strips R x [0,epsilon_1] and R x [0,epsilon_2] are non-isomorphic whenever epsilon_1 differs from epsilon_2, so the graph determines the width; Theorem 2 is the multidimensional analog for layers L(n,m,p,epsilon_1) and L(n,m,p,epsilon_2) with n >= 2 and p in (1,infinity). Theorem 3, proved in an appendix, shows that for n >= 2 and epsilon > 0 every automorphism of the unit distance graph of L(n,1,2,epsilon) = R^n x [0,epsilon] is an isometry, a Beckman-Quarles-type rigidity statement paralleling the classical 1953 theorem and Sokolov's 2023 rational analog. The introduction surveys chromatic numbers of the plane and of layers, quoting 5 <= chi(R^2) <= 7 after de Grey, the Kanel-Belov-Voronov-Cherkashin estimates for the layers R^2 x [0,epsilon]^d, and 10 <= chi(L(3,6,2,epsilon)) <= 15. For problem 508 the paper was archived from the citation sweep as structural theory of unit-distance strips; it yields no bound for the ordinary plane.

Source: https://arxiv.org/abs/2505.07799. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2505.07799), every other right reserved.

Bears on. #508: the paper studies unit distance graphs of strips and layers, the graph of the plane restricted to a strip being the object of Theorem 1; its introduction quotes 5 <= chi(R^2) <= 7 and chromatic bounds for layers from the literature. Its theorems concern isomorphisms and automorphisms of these graphs, not their chromatic numbers, and give no bound for the plane.

Results. Theorem 1 (p. 4); Theorem 2 (p. 4, with the Remark on p = infinity); Theorem 3 (p. 4, proved in the appendix on p. 25). The numbered propositions, lemmas and corollaries of Sections 3 and 4 are proof steps, summarized in the proof pointers on those pages.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.