The 2-extendability of 5-connected graphs on surfaces with large representativity

A graph with at least 2k+2 vertices is said to be k-extendable if any independent set of k edges in it extends to a perfect matching. We shall show that every 5-connected graph G of even order embedded on a closed surface F2, except the sphere, is 2-extendable if ρ(G)⩾7−2χ(F2), where ρ(G) stands for the representativity of G on F2 and χ(F2) for the Euler characteristic of F2.