Counting 3-colorings of the hex lattice

Definitions and mathematical statement

Tile the Euclidean plane by unit equilateral triangles. For each integer $n\ge1$, take the regular hexagonal patch whose six sides consist of $n$ triangle edges. Identify each side with its opposite side by translation, matching the subdivision vertices. This quotient is a triangulated torus $T_n$. Keep the resulting cellular map structure, including distinct edges if identifications produce parallel edges at a small value of $n$.

Let $H_n$ be the geometric dual of this torus triangulation: it has one vertex for each triangular face of $T_n$, and an edge across each primal edge joining its incident face vertices. Thus $H_n$ is the toroidal hexagonal lattice map associated with this specific periodic patch. There are $6n^2$ triangular faces before identifications, so $|V(H_n)|=6n^2$.

For any finite graph $G$, write $P_G(3)$ for the number of maps $c:V(G)\to\{1,2,3\}$ assigning different labels to the endpoints of every edge. Color labels are distinguished; colorings related by a global color permutation are counted separately. The problem is to prove convergence and determine the exact value of \[ \lim_{n\to\infty}P_{H_n}(3)^{1/(6n^2)}. \] This is the exponential growth factor per vertex for vertex colorings of the stated periodic hexagonal graphs. It is not an edge-coloring or face-coloring model. The specified torus identifications are part of the sequence, rather than free boundary conditions on a planar patch.

Short English statement

Find the limiting number of proper three-colorings per vertex for the specified periodic hexagonal lattice on a torus.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3234 (OPG-1764), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Counting 3-colorings of the hex lattice. Original problem and discussion. http://www.openproblemgarden.org/op/counting_3_colorings_of_the_hex_lattice.