Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 529 — #537
13.4. Bounding the Number of Edges in a Planar Graph 529
graph with v G C v H vertices, e G C e H C 1 edges, and f G C f H
1 faces. Since
.v G C v H / .e G C e H C 1/ C .f G C f H 1/
D .v G e G C f G / C .v H e H C f H / 2
D .2/ C .2/ 2 (by structural induction hypothesis)
D 2;
v e C f remains equal to 2 for the constructed embedding. That is, P.E/ also
holds in this case.
This completes the proof of the constructor cases, and the theorem follows by
structural induction.
13.4 Bounding the Number of Edges in a Planar Graph
Like Euler’s formula, the following lemmas follow by structural induction directly
from Definition 13.2.2.
Lemma 13.4.1. In a planar embedding of a connected graph, each edge occurs
once in each of two different faces, or occurs exactly twice in one face.
Lemma 13.4.2. In a planar embedding of a connected graph with at least three
vertices, each face is of length at least three.
Combining Lemmas 13.4.1 and 13.4.2 with Euler’s Formula, we can now prove
that planar graphs have a limited number of edges:
Theorem 13.4.3. Suppose a connected planar graph has v 3 vertices and e
edges. Then
e 3v 6: (13.3)
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 f faces. By Lemma 13.4.1, every edge has exactly two occurrences in the
face boundaries. So the sum of the lengths of the face boundaries is exactly 2e.
Also by Lemma 13.4.2, when v 3, each face boundary is of length at least three,
so this sum is at least 3f . This implies that
3f 2e: (13.4)