Ammann-Beenker铺砌上的哈密顿环
Hamiltonian Cycles on Ammann-Beenker Tilings
AI总结：
本研究提出构造二维Ammann-Beenker铺砌有限子图哈密顿环的简单算法，结合其离散标度对称性，得到等权旅行商、最长路径、三着色等NP完全问题的精确解，可应用于准晶体成像、催化、蛋白质折叠建模等领域。
摘要：
我们提出了一种简单算法，用于在非周期二维Ammann-Beenker（AB）铺砌的任意大有限子图集上构造哈密顿图环（即恰好访问每个顶点一次的环）。利用这一结果以及AB铺砌的离散标度对称性，我们得到了一系列其他问题的精确解，而这些问题对于一般图而言属于NP完全复杂度类。这些问题包括等权旅行商问题，其解例如可给出扫描隧道显微镜针尖对具有AB对称性的物理准晶体原子成像时可采用的最高效路径；最长路径问题，其解表明任意长度的柔性分子集合都能以数密度1吸附在AB准晶体表面，在催化领域具有潜在应用；以及三着色问题，其给出了定义在AB平面对偶图上、描述磁相互作用的q态Potts模型（q≥3）的基态，该模型可为蛋白质折叠提供有用的模型。
英文摘要：
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 ($q \ge 3$) of magnetic interactions defined on the planar dual to AB, which may provide useful models for protein folding.
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