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A domino tiling covers a region made of unit squares by nonoverlapping $1\times 2$ rectangles, each occupying two adjacent squares. Equivalently, it is a perfect matching in the graph whose vertices are squares and whose edges join squares sharing a side. This translates a geometric covering problem into graph theory.

A $2\times n$ rectangle has $T_n$ tilings satisfying $T_n=T_{n-1}+T_{n-2}$, with $T_0=T_1=1$. At the left edge, either one vertical domino leaves a $2\times (n-1)$ rectangle, or two horizontal dominoes leave a $2\times (n-2)$ rectangle. Hence $T_2=2$, $T_3=3$ and $T_4=5$. A checkerboard colouring gives a different tool: every domino covers one square of each colour, so unequal colour counts obstruct a tiling. Equal counts are necessary but need not suffice for arbitrary regions.

Robin Pemantle studies probability theory and random combinatorial structures. With Robert Burton, he developed methods for computing local probabilities in uniform spanning trees using electrical transfer impedances.^[ref]

The probability that several specified edges occur is a determinant of a small transfer matrix. This avoids enumerating all trees and relates joint edge events to Green's functions and network currents. Their work also applies these methods to domino tilings and entropy.^[ref]