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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 $$0\le n-e+f\le 2. $$

If we consider following graph $C_6+e$, and the toroidal embedding of $C_6+e$ contains $6$ vertices , $7$ edges and $2$ faces. so $n-e+f=6-7+2=1$. Obviously $0\le 1 \le 2$. enter image description here

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    $\begingroup$ The lower bound is proven in "C. Thomassen, The Jordan-Schönflies Curve Theorem and the classification of surfaces, Amer. Math. Monthly 99 (1992) 116-130." $\endgroup$ Jan 26, 2021 at 12:44
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    $\begingroup$ Thanks, I will read it. $\endgroup$
    – licheng
    Jan 26, 2021 at 14:34

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