Matching Extension Missing Vertices and Edges in Triangulations of Surfaces

Let G be a 5-connected triangulation of a surface Σ different from the sphere, and let χ = χ(Σ) be the Euler characteristic of Σ. Suppose that V0 ⊆ V (G) with |V (G) − V0| even and M and N are two matchings in G − V0 of sizes m and n respectively such that M ∩ N = ∅. It is shown that if the pairwise distance between any two elements of V0 ∪ M ∪ N is at least 5 and the face-width of the embedding of G in Σ is at least max{20m− 8χ− 23, 6}, then there is a perfect matching M0 in G−V0 containing M such that M0 ∩N = ∅.