If $G$ is a 2-connected loopless planar graph, and for each vertex $v$ we define: $f(v) = (1/2) - (1/deg(v))$, where $deg(v)$ is the degree of vertex $v$,
Show that for some region $R: \sum f(v) < 1 $, where the sum is over all vertices $v$ incident with $R$.
I'm confused about how to go about this. Some properties that could be relevant are:
- for 2-connected planar graphs, every region is bounded by a cycle
- $12 \leq \sum [6 - deg(v)]$ so $\sum deg(v) \leq 6|V(G)| - 12$
- $\sum deg(v) = 2 |E(G)| $
- every region is clearly bounded by at least 3 edges and thus has at least 3 vertices incident to it
for 2-connected graphs, every vertex has $deg(v) \geq 2$
Anyone have ideas?