Perfect Matching Complexes of Honeycomb Graphs

The perfect matching complex of a graph is the simplicial on edge set with facets corresponding to matchings graph. This paper studies complexes, $\mathcal{M}_p(H_{k \times m\times n})$, honeycomb graphs. For $k = 1$, $\mathcal{M}_p(H_{1\times n})$ contractible unless $n\geq m=2$, in which case it homotopy equivalent $(n-1)$-sphere. Also, $\mathcal{M}_p(H_{2\times 2\times 2})$ wedge two 3-spheres. proofs use discrete Morse theory.