The matching extendability of surfaces

A connected graph G having at least 2n + 2 vertices is said to be n-extendable if it contains a matching of size n and every such matching is contained in a perfect matching. M. D. Plummer posed the problem of determining the smallest integer μ(Σ) such that no graph embeddable in the surface Σ is μ(Σ)-extendable. We call μ(Σ) the matching extendability of Σ and show that if Σ is not homeomorphic to the sphere then μ(Σ) = 2 + ⌊ 4 − χ ⌋ where χ is the Euler characteristic of Σ. In particular, no projective planar graph is 3-extendable.