We develop the theory of the edge coloring of infinite lattice graphs, proving a necessary and sufficient condition for a proper edge coloring of a patch of a lattice graph to induce a proper edge coloring of the entire lattice graph by translation. This condition forms the cornerstone of a method that finds nearly minimal or minimal edge colorings of infinite lattice graphs. In case a nearly minimal edge coloring is requested, the running time is $O(\mu^2 D^4)$, where $\mu$ is the number of edges in one cell (or `basis graph') of the lattice graph and $D$ is the maximum distance between two cells so that there is an edge from within one cell to the other. In case a minimal edge coloring is requested, we lack an upper bound on the running time, which we find need not pose a limitation in practice; we use the method to minimal edge color the meshes of all $k$-uniform tilings of the plane for $k\leq 6$, while utilizing modest computational resources. We find that all these lattice graphs are Vizing class~I. Relating edge colorings to quantum circuits, our work finds direct application by offering minimal-depth quantum circuits in the areas of quantum simulation, quantum optimization, and quantum state verification.

翻译：本文发展了无限网格图边着色的理论，证明了一个网格图子图的正常边着色可通过平移诱导整个网格图正常边着色的充要条件。该条件构成了一种为无限网格图寻找近最优或最优边着色方法的核心基础。当需要近最优边着色时，算法运行时间为$O(\mu^2 D^4)$，其中$\mu$是网格图一个单元（或"基图"）中的边数，$D$是两个单元之间的最大距离（单元间存在一条边连接这两个单元）。若需最优边着色，我们缺乏运行时间的上界，但实际中这并不构成限制：利用该方法，在中等计算资源下，我们为平面上所有$k\leq 6$的$k$均匀镶嵌的网格图实现了最优边着色，发现这些网格图均属于Vizing第一类。通过将边着色与量子电路关联，本工作在量子模拟、量子优化及量子态验证领域提供了最小深度量子电路的直接应用。