Perfect Matchings and Perfect Squares

In 1961, P.W. Kasteleyn enumerated the domino tilings of a 2n × 2n chessboard. His answer was always a square or double a square (we call such a number "squarish"), but he did not provide a combinatorial explanation for this. In the present thesis, we prove by mostly combinatorial arguments that the number of matchings of a large class of graphs with 4-fold rotational symmetry is squarish; our result includes the squarishness of Kasteleyn's domino tilings as a special ease and provides a combinatorial interpretation for the square root. We then extend our result to graphs with other rotational symmetries.