Restricted matching in graphs of small genus

A graph GG with at least 2n+22n+2 vertices is said to be nn-extendable   if every set of nn disjoint edges in GG extends to (i.e., is a subset of) a perfect matching. More generally, a graph is said to have property E(m,n)E(m,n) if, for every matching MM of size mm and every matching NN of size nn in GG such that M∩N=0̸M∩N=0̸, there is a perfect matching FF in GG such that M⊆FM⊆F, but F∩N=0̸F∩N=0̸. GG is said to have property E(0,0)E(0,0) if it has a perfect matching. The study of the properties E(m,n)E(m,n) is referred to as the study of restricted matching extension.In [M. Porteous, R. Aldred, Matching extensions with prescribed and forbidden edges, Australas. J. Combin. 13 (1996) 163–174; M. Porteous, Generalizing matching extensions, M.A. Thesis, University of Otago, 1995; A. McGregor-Macdonald, The E(m,n)E(m,n) property, M.Sc. Thesis, University of Otago, 2000], Porteous and Aldred, Porteous and McGregor-Macdonald, respectively, studied the possible implications among the properties E(m,n)E(m,n) for various values of mm and nn. In an earlier paper [R.E.L. Aldred, Michael D. Plummer, On restricted matching extension in planar graphs, in: 17th British Combinatorial Conference (Canterbury 1999), Discrete Math. 231 (2001) 73–79], the present authors completely determined which of the various properties E(m,n)E(m,n) always hold, sometimes hold and never hold for graphs embedded in the plane. In the present paper, we do the same for embeddings in the projective plane, the torus and the Klein bottle.