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 thatV 0 ⊆ V ( G )with| V ( G ) − V 0 |even and M and N are two matchings in G − V 0of sizes m and n respectively such that M ∩ N = ∅ . It is shown that if the pairwise distance between any two elements ofV 0 ∪ M ∪ N is at least five and the face‐width of the embedding of G in Σ is at least max { 20 m − 8 χ − 23 , 6 } , then there is a perfect matching M 0 in G − V 0containing M such thatM 0 ∩ N = ∅ .