Let G a simple 2-connected planar graph so that all vertices are incident with the infinite region. Suppose that every bounded region of G has length 3 (so is a cycle of length 3). Let k be the number of vertices of degree 2 in G, and let r be the number of regions of G sharing no edges with the infinite region.
If |V(G)| > 3, show that
k = r + 2
I'm trying to figure out how to go about this. So, I think that these could be of help:
- every edge bounds two regions
- if the region shares no edge with the infinite region, then it only shares edges with other regions of length 3.
- 12≤∑[6−deg(v)] so ∑deg(v)≤6|V(G)|−12
for 2-connected graphs, every vertex has deg(v)≥2