Gruber P. Convex and Discrete Geometry

Corollaries of Euler’s Formula for Graphs in E 2
15 Combinatorial Theory of Convex Polytopes 263
The following consequences of Euler’s formula will be needed later.
Corollary 15.1. Let H be a planar graph in E 2 with v vertices, e edges and f countries.
If H contains no 3-cycle, then
(3) e ≤ 2v − 4.
Here equality holds if and only if H is 2-connected and each country of H has a
boundary 4-cycle.
Proof. Assume, first, that H is 2-connected. Then each edge is on the boundary of
two distinct countries of H. Since the boundary cycles of the countries are at least
4-cycles, we see that
4 f ≤ 2e,
where equality holds if and only if each country of H has a boundary 4-cycle. Since
8 = 4v − 4e + 4 f,
by Euler’s formula, addition yields (3), where equality holds if and only if each
country of H has a boundary 4-cycle.
If H is not 2-connected, there is inequality in (3) as can be shown by induction
on v by dissecting H into two or more sub-graphs which have, pairwise, at most one
vertex in common. ⊓⊔
Corollary 15.2. Let H be a planar 3-connected graph in E 2 . Then there is a vertex
incident with three edges or a country with a boundary 3-cycle.
Proof. Let v,e, f denote the numbers of vertices, edges and countries of H, respectively.
Since H is 3-connected and each country has a boundary cycle consisting of
at least 3 vertices, we may represent v and f in the form
v = v3 + v4 +··· , f = f3 + f4 +··· ,
where vk is the number of vertices which are incident with precisely k edges and fk
is the number of countries the boundary cycle of which consists of precisely k edges.
Each edge is incident with precisely two vertices and, noting that H is 3-connected,
each edge is on the boundary of precisely two countries. Hence
Euler’s formula then yields,
2e = 3v3 + 4v4 +···=3 f3 + 4 f4 +···
8 = 4v − 4e + 4 f = 4v3 + 4v4 + 4v3 +···−3v3 − 4v4 − 5v5 −···
− 3 f3 − 4 f4 − 5 f5 −···+4 f3 + 4 f4 + 4 f5 +···
= v3 − v5 − 2v6 −···+ f3 − f5 − 2 f6 −···≤v3 + f3. ⊓⊔