The closed-form formula for the number of domino tilings of a $2n\times 2n$ board is known to be
$$\prod_{j=1}^{n}\prod_{k=1}^{n}\Big{(}4\cos^2\frac{\pi j}{2n+1} + 4\cos^2\frac{\pi k}{2n+1}\Big{)}.$$
(Explicitly, A004003 in OEIS). Each such tiling can be colored in $4$ colors. With respect to this property, tilings naturally split into $3$ disjoint classes:
- $\mathcal{F}$: those which cannot be colored in $3$ colors (for $n=2$, $|\mathcal{F}|=6$);
- $\mathcal{M}$: non-uniquely $3$-colorable tilings (for $n=2$, $|\mathcal{M}|=30$);
- $\mathcal{U}$: uniquely $3$-colorable tilings (for $n=2$, $|\mathcal{U}|=0$);.
The first two classes have a local structure in the following sense. Any tiling containing a certain $3\times 4$ domino pattern $\mathscr{O}$ (see the top-left tiling where it is marked in bold) belong to the class $\mathcal{F}$. Analogous phenomenon holds for the class $\mathcal{M}$ (the bottom picture is self-explanatory). On the other hand, class $\mathcal{U}$ is the most intriguing, since the unique 3-colorability is a global feature of an adjacency graph of the tiling.
My questions are as follows:
Can one write down a closed-form formula for the number of tilings in the class $\mathcal{U}$? Or, at least, a program to enumerate them?
Is it true that any tiling in the class $\mathcal{F}$ contains a subtiling $\mathscr{O}$ (or, relaxing this condition, one of subtilings from a given finite collection)?
There are many papers on unique 3-colorability of (planar or non-planar) graphs, but I could not find any reference related to adjacency graphs of tilings.