TITLE: In a graph, can an edge be in less than 2 faces?
QUESTION [12 upvotes]: In the proof that for every conective plane graph on $n$ vertices and $m$ edges $m\leq 3(n-2)$ I encountered the statement: $\Sigma_{f\in F} f\leq 2m$, and the explanation was that every edge is in at most 2 faces.
Can someone explain why that is not exactly 2 faces?

REPLY [18 votes]: Consider any tree - it has no cycles, hence its graph has just one (unbounded) face meaning each edge is a part of just a single face. More generally, this is true of any edge which is not part of a cycle.

REPLY [12 votes]: Consider the following hastily made graph
            .----.

consisting of $2$ vertices and one edge.