Distance-restricted matching extension in triangulations of the torus and the Klein bottle

A graph G with at least 2m + 2 edges is said to be distance d m-extendable if for any matching M in G with m edges in which the edges lie pair-wise distance at least d, there exists a perfect matching in G containing M. In a previous paper, Aldred and Plummer proved that every 5-connected triangulation of the plane or the projective plane of even order is distance 5 m-extendable for any m. In this paper we prove that the same conclusion holds for every triangulation of the torus or the Klein bottle.