0 On the restricted matching of graphs in surfaces ∗

A connected graph G with at least 2m + 2n + 2 vertices is said to have property E(m,n) if, for any two disjoint matchings M and N of size m and n respectively, G has a perfect matching F such that M ⊆ F and N ∩ F = ∅. In particular, a graph with E(m, 0) is m-extendable. Let μ(Σ) be the smallest integer k such that no graphs embedded on a surface Σ are k-extendable. Aldred and Plummer have proved that no graphs embedded on the surfaces Σ such as the sphere, the projective plane, the torus, and the Klein bottle are E(μ(Σ)− 1, 1). In this paper, we show that this result always holds for any surface. Furthermore, we obtain that if a graph G embedded on a surface has sufficiently many vertices, then G has no property E(k − 1, 1) for each integer k ≥ 4, which implies that G is not k-extendable. In the case of k = 4, we get immediately a main result that Aldred et al. recently obtained.