# Counting 3-colorings of the hex lattice

**Problem**Find .

We'll begin by putting in place the necessary notation. Let be the regular triangular tiling of the plane. For every there is a regular map which triangulates the torus, denoted , which may be obtained from a regular hexagonal piece of of side-length by identifying points on opposite edges of this hexagon. Let be the dual of (on the torus). Then is a regular map on the torus - a hexagonal tiling. One last definition: for any graph and any positive integer we let denote the number of proper -coloring of .

A famous theorem of Lieb [L] shows that where denotes the quadrangulation of the torus. This theorem is usually stated in terms of Eulerian orientations, and is of interest to physicists as the constant (called Lieb's Ice Constant) determines the "residual entropy for square ice".

Thomassen proved that every planar graph with girth has exponentially many proper 3-colorings. More precisely, he showed that . This gives a lower bound on the limit in the above problem (assuming it exists).

## Bibliography

[L] E. H. Lieb, Exact Solution of the Problem of the Entropy of Two-Dimensional Ice. Phys. Rev. Lett. 18, 692-694, 1967.

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