How to find cubic non-snarks where the $\min(f_k)>6$ on surfaces with $\chi<0$? Henning once told me that,

[i]t follows from the Euler characteristic of the plane that the average face degree of a 3-regular planar graph with $F$ faces is $6-12/F$, which means that every 3-regular planar graph has at least one face with degree $5$ or lower.

I tried to understand and extend this and got the following:
Given a $k$-regular graph. Summing over the face degrees $f_n$ gives twice the number of edges $E$ and this is $k$ times the number of vertices $V$:
$$
\sum f_n = 2E =kV \tag{1}
$$
Further we have Euler's formula saying
$$
V+F = E +\chi, 
$$
where  $\chi$ is Euler's characteristic of the surface where the graph lives on. Again we insert $E=\frac k2V$ and get:
$$
F=\left( \frac k2 -1 \right)V+\chi\\
V=\frac{F-\chi}{\frac k2 -1}. \tag{2}
$$
Dividing $(1)$ by $F$ and inserting $(2)$ gives:
$$
\frac{\sum f_n}{F}= \frac{k(F-\chi)}{(\frac k2 -1)F}=\frac {2k}{k -2} \left( 1-\frac{\chi}{F}\right) \tag{3}
$$
or
$$
\sum f_n=\frac {2k}{k -2} \big( F-\chi\big). \tag{3$^\ast$}
$$
Plug in $k=3$ and $\chi=2$ 
(the characteristic of the plane), we get back Henning's formula, but when e.g. $\chi=-2$, so the surface we draw could be a double torus, we get the average degree to be:
$$
6 \left( 1+\frac{2}{F}\right)
$$
How to find cubic graphs where the $\min(f_k)>6$ on surfaces with $\chi<0$?
EDIT The graph should not be a snark.

$\hskip2.5in$
This genus-$2$ regular map, shown to the right, has six octagonal faces, of which three meet at each of its $16$ vertices. It has $24$ edges, and a Euler characteristic of $-2$.
Since it's bipartite there is a $3$-edge-coloring, so it's not a snark...
