Edge coloring lattice graphs

TL;DR

This work addresses edge coloring on infinite lattice graphs by introducing patches of a lattice graph and a wrapping technique that reduces the problem to finite, manageable subgraphs. The authors prove that a wrapped patch yields a proper coloring of the lattice if and only if the patch is self-loop-free, enabling a constructive method to obtain type-I (minimal) or type-II colorings; a worst-case running time bound of $O(D^4\mu^2)$ is established for a practical variant. The method is implemented and applied to a broad suite of lattice graphs, yielding type-I colorings for all Archimedean, Laves, and $k$-uniform tilings with $k\le 6$, and demonstrating class I in these cases; explicit class-II examples are also presented to delineate limitations. The practical impact lies in providing minimal-depth quantum circuits for quantum simulation, optimization, and verification, by mapping edges to gate layers in depth-optimal quantum computations.

Abstract

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 $O(μ^2 D^4)$, where $μ$ 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.

Figures (7)

• Figure 1: Lattice graphs. A basis graph $B$ (bold vertices and edges), containing $\mu$ edges, induces a lattice graph $\mathcal{G}$ by the union of infinitely many translated copies of $B$. A patch $P_{n,m}$ of $\mathcal{G}$ is formed by the union of $n \times m$ copies of $B$. (In the figure, $n=m=3$, and $P_{n,m}$ is formed by all vertices and edges shown). Lattice graphs need not be planar, as symbolized by the slash through the edge between vertices $s$ and $s'$. The seeds $\mathscr S$ are the vertices inside the unit cell [parallelogram labeled $(0,0)$]. An edge-colored patch $C(P_{n,m})$ induces an edge coloring $\mathcal{C}$ of $\mathcal{G}$ by the union of infinitely many translated copies of $C(P_{n,m })$ (not illustrated here).
• Figure 2: Main stages of Method \ref{['met:edge_coloring']}. The preprocessing stage is to define the lattice graph by its basis graph (Fig. \ref{['fig:lattice_graph']}). Then, stages 1--3 are repeated with increasing patch size until a patch is found at stage 3 that is self-loop-free and permits a type-$t$ ($t=1,2,3$) edge coloring. At stage 4, this patch is unwrapped, retaining its edge coloring. As a postprocessing stage (stage 5), the infinite lattice graph is generated purely abstractly, or a finite patch is generated explicitly.
• Figure 3: Minimal edge colorings of the 11 Archimedean lattice graphs. The coloring basis graph, that is, the coloring pattern that needs to be repeated to color the entire lattice graph, is depicted with bold lines, both solid and dashed. It always contains the basis graph of the uncolored lattice graph (bold, solid lines only). Below the lattice graph follow various names of that graph. The first name is in the notation of Grünbaum and Shephard grunbaum2016tilings. Starting at any vertex, it lists the number of vertices of each of the adjacent polygons in cyclic order. Any repetitions are abbreviated by a superscript, e.g., $(6^3)=(6,6,6)$. This name is followed by various other (possibly nonsystematic) names, whereas the last name is the one given by Conway et al. conway2008symmetries.
• Figure 4: Minimal edge colorings of all Laves tilings that are not already in Fig. \ref{['fig:archimedean']}. The data are presented as in Fig. \ref{['fig:archimedean']}, except for the first name, which is now the name of the Laves lattice by Grünbaum and Shephard grunbaum2016tilings. Taking any tile, it lists (in square brackets) the degree of the tile's vertices in the lattice. The dual of $(x)$ is $[x]$.
• Figure 5: Minimal edge coloring of a 6-uniform geometrical lattice graph from Refs. galebach2023n_uniformgalebach_json.

Theorems & Definitions (19)

• Definition 1: Basis graph
• Definition 2: Lattice graph
• Definition 3: Patch
• Theorem 1
• Lemma 1
• Theorem 2
• Proposition 1
• proof
• Lemma 2
• proof