Edge coloring lattice graphs
<|reference_start|>Edge coloring lattice graphs: 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.<|reference_end|>
@article{kattemölle2024edge,
    title={Edge coloring lattice graphs},
    author={Joris Kattem"olle},
    journal={arXiv preprint arXiv:2402.08752},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.08752},
    primaryClass={quant-ph cs.DM math.CO}
}