Lemma 5.8.4. In a planar embedding of a connected graph, each edge is traversed once by each of two different regions, or is traversed exactly twice by one region.
Lemma 5.8.5. In a planar embedding of a connected graph with at least three vertices, each region is of length at least three.
Suppose a connected planar graph has $v\geq 3$ vertices and $e$ edges. Then
$$e\leq 3v−6.$$
Proof. By definition, a connected graph is planar iff it has a planar embedding. So suppose a connected graph with v vertices and e edges has a planar embedding with r region. By Lemma 5.8.4, every edge is traversed exactly twice by the region boundaries. So the sum of the lengths of the region boundaries is exactly 2e. Also by Lemma 5.8.5, when v≥3, each region boundary is of length at least three, so this sum is at least 3r . This implies that
3r≤2e
But r=e−v+2 by Euler’s formula, and substituting into the above gives
3(e−v+2)≤2e
=e−3v+6≤2e
=e≤3v−6
My question is that "sum of the lengths of the region boundaries is exactly 2e" doesn't quite makes sense. How can I understand the "2e" part according to number of edges, cause it looks like 2e should be total/maximum number of regions if we have e edges in a graph and each edge contributes to 2 regions.