On the matching complexes of categorical product of path graphs

The matching complex M (G) of a graph G is a simplicial complex whose simplices are matchings in G. These complexes appears in various places and found applications in many areas of mathematics including; discrete geometry, representation theory, combinatorics, etc. In this article, we consider the matching complexes of categorical product Pₙ Pₘ of path graphs Pₙ and Pₘ. For m = 1, Pₙ Pₘ is a discrete graph and therefore its matching complex is the void complex. For m = 2, M (Pₙ Pₘ) has been proved to be homotopy equivalent to a wedge of spheres by Kozlov. We show that for n 2 and 3 m 5, the matching complex of Pₙ Pₘ is homotopy equivalent to a wedge of spheres. For m =3, we give a closed form formula for the number and dimension of spheres appearing in the wedge. Further, for m \4, 5\, we give minimum and maximum dimension of spheres appearing in the wedge in the homotopy type of M (Pₙ Pₘ).