Homology of matching complexes of $3\times n$ grid graphs

For a finite simple graph $G$, the matching complex $M(G)$ is the simplicial complex whose vertex set is the edge set of $G$ and whose simplices are all the matchings in $G$. The topology of the matching complex of the $m\times n$ grid graph $G_{m\times n}$ is known only for $m = 1,2$, in which cases it is homotopy equivalent to a wedge of spheres. In this article, we study the matching complex $M(G_{3 \times n})$. We prove that for $n\ge2$, its reduced homology vanishes in dimensions $i \leq n-2$ and in top dimension, while $\tilde{H}_{n-1}(M(G_{3\times n}))\neq 0$. We also show that $M(G_{3 \times n})$ is simply connected for $n \geq 3$. Consequently, the topological connectivity of $M(G_{3\times n})$ is $n-2$.