mortal sleet
I am working on a problem about planar graphs and trying to understand if two statements can both be true at the same time. The problem states that for any planar graph with at least 3 or more vertices, the number of edges is limited by a known inequality.
The first statement makes sense to me because Euler's formula and the constraints on the number of edges in a planar graph mean that there must always be at least one vertex with degree 5 or less.
The second statement is where I get confused. While it's true that no node in a planar graph can have a degree greater than 12, I am not sure if it is always guaranteed that at least half of the nodes have a degree of 12 or less.
So my question is: Is it possible for both (1) and (2) to be true at the same time? Or does (2) fail in some cases, making (1) the only valid statement?
fathom phoenix