Hamiltonian Cycles on Ammann-Beenker Tilings,Physical Review X

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Hamiltonian Cycles on Ammann-Beenker Tilings
Physical Review X ( IF 11.6 ) Pub Date : 2024-07-10 , DOI: 10.1103/physrevx.14.031005
Shobhna Singh ₁ , Jerome Lloyd ₂ , Felix Flicker ₃

Affiliation

We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems which lie in the complexity class NP-complete for general graphs. These include the equal-weight traveling salesperson problem, providing, for example, the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals with AB symmetries; the longest path problem, whose solution demonstrates that collections of flexible molecules of any length can adsorb onto AB quasicrystal surfaces at density one, with possible applications to catalysis; and the three-coloring problem, giving ground states for the

q

-state Potts model (

q \geq 3

) of magnetic interactions defined on the planar dual to AB, which may provide useful models for protein folding.

中文翻译：

Ammann-Beenker 平铺上的哈密顿循环

我们提供了一种简单的算法，用于在非周期二维 Ammann-Beenker (AB) 平铺的一组任意大的有限子图上构建哈密顿图循环（仅访问每个顶点一次）。利用这个结果以及 AB 平铺的离散尺度对称性，我们找到了一般图的复杂性类别 NP 完全中的一系列其他问题的精确解决方案。其中包括等重旅行推销员问题，例如，提供扫描隧道显微镜尖端对具有 AB 对称性的物理准晶体原子进行成像的最有效路线；最长路径问题，其解决方案表明任意长度的柔性分子的集合可以以密度1吸附到AB准晶表面上，并可能应用于催化；和三色问题，给出了在平面对偶 AB 上定义的磁相互作用的