Abstract
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.