Proximity thresholds for matching extension in the torus and Klein bottle

A graph $G$ is said to have the property $E_d(m,n)$ if, given any two disjoint matchings $M$ and $N$ such that the edges within $M$ are pair-wise distance at least $d$ from each other as are the edges in $N$, there is a perfect matching $F$ in $G$ such that $M?F$ and $F\cap N=0?$. This property has been previously studied for planar triangulations as well as projective planar triangulations. Here this study is extended to triangulations of the torus and Klein bottle.