One and Seven-Eighths Divisibility Problems of Propp
Aidan Dunkelberg
Source abstract
We provide one full solution, and another nearly complete solution, to two problems from Jim Propp's 1999 article "Enumeration of matchings; problems and progress" (namely, Problems 30 and 31) involving divisibility properties of the number of matchings of two non-bipartite triangular graphs. We extend our method of proof of the second problem to show that the number of matchings of any graph composed of tetrahedral cells, such that the number of cells is suitably few relative to the number of vertices, is divisible by a power of 3; in particular, we then exhibit a collection of non-planar graphs whose number of matchings is divisible by 3.
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