MATH 682
Notes
Combinatorics and Graph Theory II
1
Planar graphs
1.1
More fun with faces
We developed this idea of a “face” of a planar projection of a graph
G
, which motivated the very
useful result known as Euler’s Formula: if a planar projection of a connected graph has
v
vertices,
f
faces, and
e
edges, then
v
+
f

e
= 2. There is one useful immediate consequence of this theorem:
Corollary 1.
The number of faces of a planar projection connected graph
G
does not depend upon
the choice of projection.
We can develop several other properties of faces besides simply their number.
Two interesting
qualities of faces are their adjacencies and boundary lengths.
Definition 1.
If the edges bounding a face
F
form a closed walk of length
i
, the face itself is said
to have
length
i
.
Note that edges separating the face from other faces will be counted once, while edges that jut into
the face will be counted twice. This brings us to our second concept; namely, that two faces might
share an edge.
Definition 2.
If the edge
e
lies on the boundary of two separate faces
f
1
and
f
2
, then
f
1
and
f

2
are said to be adjacent.
Definition 3.
The
dual graph
of a planar projection of
G
is a graph
H
whose vertices are the faces
of
V
, in which
f
1
and
f
2
are adjacent in
H
if they are adjacent in
V
.
The dual and face lengths are useful visualization tools, but have one grievous flaw: they aren’t
actually uniquely defined by the graph
G
, but only by its planar projection! Below are two planar
projections of the same graph, with quite different duals ansd facelengths.
For that reason, looking at the faces of
G
can be inherently dangerous if we treat them as if they
were uniquely defined. However, to a certain extent we can make statements about them which are
true regardless of projection. For instance:
Proposition 1.
If a planar graph
G
has faces
f
1
, . . . , f
k
with respective lengths
‘
1
, . . . , ‘
k
, then
∑
k
i
=1
‘
i
= 2
k
G
k
.
Proof.
Every edge is either a boundary between two faces, in which case it lies on the boundary
walk of each face, contributing 1 towards the length of each and 2 towards the total, or lies wholly
surrounded by a single face, in which case it appears twice on the boundary walk for a single
face, contributing twice towards its length, and thus twice towards the total. Thus, each edge is
accounted for twice in a census of all faceboundaries, so the above sum is twice the number of
edges in
G
.
Proposition 2.
In a planar connected graph
G
with 3 or more vertices, every face has length at
least 3.
Page 1 of 8
February 18, 2010
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
MATH 682
Notes
Combinatorics and Graph Theory II
Proof.
If
G
is acyclic, it has a single face with boundary demonstrably of length 2

G

>
3. Oth
erwise, every face has a nontrivial walk on its boundary, which could be reduced to a cycle by
ignoring the edges which jut into the face.
Since cycles have length of at least 3, the boundary
walks have length of at least 3.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 WILDSTROM
 Combinatorics, Graph Theory, Planar graph, Graph coloring, Graph Theory II

Click to edit the document details