On the matching extendability of graphs in surfaces

A graph G with at least 2n + 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 X, 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) over bar >= 2, there are infinitely many graphs of non-orientable genus (g) over bar 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. (c) 2007 Elsevier Inc. All rights reserved.