On the restricted matching extension of graphs in surfaces

A connected graph G with at least 2 m + 2 n + 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 ) .