Selfmade revision notes for Module CS1231 Discrete Structure
CS1231 Graph Theory Summary Notes YANG LU 1
CS1231 Discrete Structure
Chapter 11 Graph Theory
Summary Notes
Contents
Section 1 Terminology
Undirected Graph
Directed Graph
Complete Graphs
Complete Bipartite Graph
Subgraphs
Vertex Degree
The Handshake Theorem
Section 2 Connectivity
Section 3 Euler Tour
Section 4 Hamilton Cycles
Section 5 Graph Representation
Last Edit: 18
th
October 2014
Section 1 Terminology
Undirected Graph
1.
A
pseudograph
=,,
consists of
a nonempty
vertex
set
of vertices
an
edge
set
of edges
an
incidence function
: →,  ,∈
.
[NOTE: vertex set
must be nonempty]
2.
Edges are
undirected
in an undirected graph.
3.
An edge
e
is a
loop
if
ⅇ=,=
for some
∈
.
4.
Two distinct edges
ⅇ
and
ⅇ
are
multiple
(or parallel) edges if
ⅇ
=
ⅇ
.
5.
A
simple graph
is a pseudograph with no loops and parallel edges
6.
A
multigraph
is a pseudograph with no loops.
7.
An edge
e
is
incident
with vertices
u
and
v
(vice versa) if
ⅇ=,
. The edge
e
is said to
connect
its endpoints
u
and
v
.
8.
Two vertices are
adjacent
(or neighbours) if they are incident with a common edge.
9.
Two edges are called
adjacent
if they are incident with a common vertex.
CS1231 Graph Theory Summary Notes YANG LU 2
Directed Graph
10.
A
directed multigraph
=,,
consists of
a nonempty
vertex
set
of vertices
an
edge
set
of edges
an
incidence function
: →, ,∈
[NOTE: difference with undirected graph here is the use of notation of round brackets instead of curly brackets
ordered pair is used
indicate the importance of sequence]
11.
The directed edge
e
start at
initial vertex u
and end at
terminal
(or end)
vertex
v
if
ⅇ=,
.
12.
A directed edge
e
is a
loop
if
ⅇ=,
for some
∈
.
13.
The directed edges
e
1
and
e
2
are
multiple directed edges
if
ⅇ
=
ⅇ
.
14.
A
simple directed graph
is a directed multigraph with no loops and multiple directed edges.
Complete Graphs
15.
A
complete graph
on
n
vertices, denoted by
K
n
, is a simple graph in which every two distinct vertices area adjacent. Number of edges in complete graph
Kn
=

= 2=
−
Complete Bipartite Graph
16.
A
complete
bipartite graph
on
(m,n)
vertices, denoted b
K
m,n
, is a simple graph with
(
,
)=
,
,...,
∪
,
…
(
,
)
=
{ {
′
} ⅈ=1…;=1…
Number of edges between
u
vertices on the top or bottom =
(
,
)=×
Subgraphs
17.
A graph
H
is a subgraph of a graph
G
if
⊆
⊆
∀ⅇ∈ (
ⅇ=
ⅇ)
Vertex Degree
18.
The
degree
of a vertex
v
in an undirected graph
G
is the number if edges incident with
v
, each loop counting as 2 edges. A vertex of degree 0 is
isolated
.
CS1231 Graph Theory Summary Notes YANG LU 3
The Handshake Theorem
THEOREM 1
Let G be an undirected graph. Then
= 2
∈
COROLLARY 1
Sum of degrees of all vertices of G is even.
PROOF
: Combinational Proof 1)
List each edge and label its endpoints with the vertex names. 2)
The number of times each vertex name is used is the vertex degree. 3)
All vertex names appear
2
times, by STEP 1. 4)
This is also the sum of all the vertex degrees, by STEP 2. Q.E.D.
COROLLARY 2
In an undirected graph, the number of vertices of odd degree is even.
PROOF
: Direct Proof 1)
Let
V
1
and
V
2
be the sets of vertices of odd and even degree in
G
, respectively. 2)
Therefore,
∈
+
∈
=
∈
3)
Therefore RHS is even, by Theorem 1. 4)
Therefore LHS is even by STEP 3. 5)
∑
∈
is even. 6)
Therefore
∑
=
∈
∈
1+ ∑1
∈
and

=∑1
∈
,


is even. Q.E.D.
CS1231 Graph Theory Summary Notes YANG LU 4
Section 2 Connectivity
1.
A
walk
of length n in an undirected graph
G
is a finite alternating sequence of vertices and edges of
G
such that,
ⅇ
ⅇ
…
−
ⅇ
ⅇ
connects its endpoints
−
and
for
I
= 1, …
n
vertices
v
o
and
v
n
are the
srcin
and
terminus
respectively
v
1
… v
n1
are
internal vertices
if in a simple graph a walk can be specified uniquely by its vertex sequence 2.
A
trail
is a walk with distinct edges. [no repeated edges] 3.
A
path
is a trail with distinct vertice. [no repeated vertices except for srcin or terminus] 4.
An undirected graph
G
is
connected
if there is a walk between every pair of distinct vertices of
G
. 5.
An undirected graph
H
is a
connected component
of the undirected graph
G
if
H
is a subgraph of
G,
H
is connected,
No connected subgraph of
G
has
H
as a proper subgraph. [Contains maximum number of edges]
THEOREM 2
There is a path between every pair of distinct vertices of a connected undirected graph
PROOF
: Direct Proof 1)
Let
u
and
v
be 2 distinct vertices of
G
. There is at least one walk between
u
and
v
, by definition. 2)
Choose the walk of least length. 3)
This walk of least length is a path.
PROOF
: Contrapositive Proof 1)
Let
,
…
be the vertex sequence of a walk between distinct vertices
v
0
and
v
n
. 2)
By definition, this walk exists because the graph is connected. If the walk is not a path, then
=
for some
i
and
j
with
0≤ⅈ≤
. 3)
Therefore, there is a walk from
to
of shorter length with vertex sequence
…
−
…
obtained by deleting the edges corresponding to the vertex sequence
…
−
.