We shall consider only simple graphs. Let $G$ be a graph 2-cell embedded in a surface having Euler characteristic $\chi$. Let $\delta(G)$ be the minimum degree of $G$. Define
$\delta_{\chi} = max \{\delta (G)$ : a graph $G$ is 2-cell embedded in a surface with Euler characteristic $\chi \}$.
I know that $\delta_2 =5, \delta_1 = 5, \delta_0 =6, \delta_{-1} =6$.
Is there formula for $\delta_{\chi}$?