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

A graph with at least 2 k + 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 F 2 , except the sphere, is 2-extendable if ρ ( G ) ⩾ 7 − 2 χ ( F 2 ) , where ρ ( G ) stands for the representativity of G on F 2 and χ ( F 2 ) for the Euler characteristic of F 2 .