Solving the dimer problem on Apollonian gasket

Abstract

For any three circles in the plane where each circle is tangent to the other two, the Descartes' theorem yields the existence of a fourth circle tangent to the starting three. Continuing this process by adding a new circle between any three tangent circles leads to Apollonian packings. The fractal structures resulting from infinite continuation of such processes are known as Apollonian gaskets. Close-packed dimer configurations on such structures are well modeled by perfect matchings in the corresponding graphs. We consider Apollonian gaskets for several types of initial configurations and present explicit expressions for the number of perfect matchings in such graphs.