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).