On the $d$-transversal number of cylindrical and toroidal grids

For a positive integer $d$, a $d$-transversal set of a graph $G$ is an edge subset $T\subseteq E(G)$ such that $|T\cap M|\geq d$ for every maximum matching $M$ of $G$. The $d$-transversal number of $G$, denoted by $\tau_d(G)$, is the minimum cardinality of a $d$-transversal set in $G$. It is NP-complete to determine the $d$-transversal number of a bipartite graph for any fixed $d\geq 1$. Ries et al. (Discrete Math. 310 (2010) 132-146) established the $d$-transversal number of rectangular grids $P_m\square P_n$. In this paper, we consider cylindrical grids $P_m\square C_n$ and toroidal grids $C_m\square C_n$. We derive explicit expressions for the $d$-transversal numbers of $P_m\square C_n$ for $m\geq 1$ and even $n\geq 4$, or even $m\geq 2$ and $n=3$, and of $C_m\square C_n$ with even order, for $1\leq d\leq \frac{mn}{2}$. For the other cases we obtain explicit expressions or bounds for their $d$-transversal numbers.