Proximity thresholds for matching extension in the torus and Klein bottle

A graph GG is said to have the property Ed(m,n)Ed(m,n) if, given any two disjoint matchings MM and NN such that the edges within MM are pair-wise distance at least dd from each other as are the edges in NN, there is a perfect matching FF in GG such that M⊆FM⊆F and F∩N=0̸F∩N=0̸. 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.