# What is Euler's Formula when 2­-cell embedding condition was removed?

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 $$0\le 1 \le 2$$.

• 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." Jan 26, 2021 at 12:44
• Thanks, I will read it. Jan 26, 2021 at 14:34