Minimum degree in any triangle free graph
$\delta =4$
by using Euler's formula
e-n+(k+1) =r {k=#f connected components}
e-10+2=r
=> e-8=r
As, min degree of region is 4
$4r\leq 2e$
=> 4(e-8) $\leq$ 2e
=> 4e-32 $\leq$ 2e
=> e $\leq$16
For a triangle free graph, every face has degree $\ge$ 4. Therefore $2e \le 4f \implies f \ge \frac{e}{2}$ (This is because $\sum deg(f_i) = 2e $). We know from Euler's formulae that $n-e+f=2$. So, $f=2+e-n \implies \frac{e}{2} \ge 2+e-n \implies 2(n-2) \ge e$. Here $n=10$. Therefore, $e\le 16$.