  The problem of map enumeration concerns counting connected spatial graphs,
with a specified number $j$ of vertices, that can be embedded in a compact
surface of genus $g$ in such a way that its complement yields a cellular
decomposition of the surface. As such this problem lies at the cross-roads of
combinatorial studies in low dimensional topology and graph theory. The
determination of explicit formulae for map counts, in terms of closed classical
combinatorial functions of $g$ and $j$ as opposed to a recursive prescription,
has been a long-standing problem with explicit results known only for very low
values of $g$. In this paper we derive closed-form expressions for counts of
maps with an arbitrary number of even-valent vertices, embedded in surfaces of
arbitrary genus. In particular, we exhibit a number of higher genus examples
for 4-valent maps that have not appeared prior in the literature.
