It is well known that Euler's Formula for genus $\gamma$: For every 2-cell embedding of a graph on a surface with genus $\gamma$, the numbers of vertices, edges, and faces satisfy $n-e+f=2-2\gamma$.
My quetion: If the 2-cell embedding condition was removed,what will we get? Does following inequality hold true in general genus $\gamma$? $$2-2\gamma\le n-e+f\le 2-2(\gamma-1)=2 \gamma$$ For example: We know a toroidal graph is a graph that can be embedded on a torus. So maybe for embedding of any toroidal graph, we would get $$0\le n-e+f\le 2. $$