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 subset of F and N boolean AND F = empty set. Let mu(Sigma) be the smallest integer k such that no graphs embedded in the surface Sigma 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(mu(Sigma) - 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). (C) 2012 Elsevier Ltd. All rights reserved.