If I describe a surface of genus $g$ using a cellularly embedded graph with $n$ vertices, can I immediately conclude that $g = O(n)$ and thus the number of edges is linear in $n$? .
Also the other direction is not clear to me, for example every platonic solid is homeomorphic to the 2-sphere, but they have different number of vertices. So is there a theorem which states that if I have a surface of genus $g$ then I need at least $f(g)$ many vertices for a polygonal schema or a cellularly embedded graph to represent the surface ?
Thank you .