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 .