On the matching extendability of graphs in surfaces

A graph G with at least 2 n + 2 vertices is said to be n-extendable if every matching of size n in G extends to a perfect matching. It is shown that (1) if a graph is embedded on a surface of Euler characteristic χ, and the number of vertices in G is large enough, the graph is not 4-extendable; (2) given g > 0 , there are infinitely many graphs of orientable genus g which are 3-extendable, and given g ¯ ⩾ 2 , there are infinitely many graphs of non-orientable genus g ¯ which are 3-extendable; and (3) if G is a 5-connected triangulation with an even number of vertices which has genus g > 0 and sufficiently large representativity, then it is 2-extendable.