Given a cubic planar hamiltonian graph with $F$ faces. Let $a_k$ be the number of face of degree $k$ inside and $b_k$ outside the Hamilton cycle. We have the following:
- $\sum \limits_k \left(a_k+b_k\right)k = 6(F-2)$ (due to Euler; see $(3^\ast)$ here)
- $\sum \limits_k \left(a_k-b_k\right)(k-2)=0$ (Grinberg's Theorem)
Are these two all of this type or are there more?