Abstrakt:  Graph $G$ is said to be equimatchable, if every matching in $G$ extends to (i.e., is a subset of) a maximum matching. In the paper
[K. Kawarabayashi and M. D. Plummer. Bounding the size of
equimatchable graphs of fixed genus. Graphs and Combinatorics, 25(1):9199, 2009.] it is showed that for any fixed $g$, there are only finitely many $3$connected equimatchable graphs $G$ embeddable in the surface of genus $g$ with the property that either $G$ is nonbipartite or the embedding has representativity at least three. The proof is based on a result that the maximum size of such a graph is at most $c\cdot g^{3/2}$, where $c$ is a constant. In this thesis we show that the upper bound on the number of vertices of a $2$connected, nonbipartite, equimatchable graph embeddable in the surface of genus $g$ is between $5\sqrt{g} + 6$ and $4\sqrt{g} + 17$ for any $g \leq 2$, between $5\sqrt{g} + 6$ and $12\sqrt{g} + 5$ for any $g\geq 3$, and between $5\sqrt{g}+6$ and $8\sqrt{g} + 5$ for $g\geq 63$. Our methods are based on and refine the concept of isolating matchings used in the aforementioned paper.
Moreover, we provide additional results concerning the structure of factorcritical equimatchable graphs and graphs embeddable in a fixed surface.

