On the restricted matching extension of graphs on the torus and the Klein bottle

Aldred and Plummer proved that every 6-connected even graph minimally embedded on the torus or the Klein bottle is E(1,n)(n≤3) and E(0,n)(n≤5) [R.E.L. Aldred, M.D. Plummer, Restricted matching in graphs of small genus, Discrete Math. 308 (2008) 5907-5921]. In this paper, we can remove the upper bounds on n by showing that every even 6-regular graph G embedded on the torus or the Klein bottle has property E(1,n−1) and E(0,n) for arbitrary n≤|V(G)|2−1.