On the restricted matching extension 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 sizes m and n respectively, G has a perfect matching F such that M⊆F and N∩F=0̸. Let μ(Σ) be the smallest integer k such that no graphs embedded in the surface Σ are k-extendable. It has been shown that no graphs embedded in some scattered surfaces as the sphere, projective plane, torus and Klein bottle are E(μ(Σ)−1,1). In this paper, we show that this result holds for all surfaces. Furthermore, we obtain that for each integer k≥4, if a graph G embedded in a surface has too many vertices, then G does not have property E(k−1,1).