MAT 2720
Discrete Mathematics
Section 8.2
Paths and Cycles
http://myhome.spu.edu/lauw
Goals

Paths and Cycles
• Definitions and Examples
• More Definitions
Definitions
v0
vn
Definitions
vn1 en
v0
vn
e1 v1
e2
v2
e3
v3
Definitions
vn1
v0
vn
v1
v3
v2
 v0 , v1, v2 ,
, vn 
Example 1
(a) Write down a path from b to e with
length 4.
Example 1
(b) Write down a path from b to e with
length 5.
Example 1
(c) Write down a path from b to e with
length 6.
Definitions
v
w
v
w
Example 2
The graph is not connected because …
a
b
c
e
d
f
Definitions
v
v
e
w
e
w
Definitions
v
e
v
v
e
w
e
w
Definitions
(b) V , E is a graph.
v
e
w
Example 3
How many subgraphs are there with 3
edges?
a
b
c
e
f
Definitions
v
Definitions
v
Connected Graph & Component
What can we say about the components of
a graph if it is connected?
v
Connected Graph & Component
What can we say about the graph if it has
exactly one component?
v
Theorem
A graph is connected if and only if it has
exactly one component
v
Definitions
w
v
u
Definitions
b
a
v
v
u
c
u
w
x
w
x
Definitions
b
a
v
v
u
c
u
w
x
w
x
Definitions
The degree of a vertex v, denoted by (v),
is the number of edges incident on v
Definitions
The degree of a vertex v, denoted by (v),
is the number of edges incident on v
e
b
a
c
f
d
w
g
u
v
h
 (a)   (b)   (c)   (d )   (e)   ( f )   ( g )   (h) 
 (u )   (v)   ( w) 
The Königsberg bridge problem

Euler (1736)

Is it possible to cross all seven bridges just once
and return to the starting point?
The Königsberg bridge problem

Edges represent bridges and each
vertex represents a region.
The Königsberg bridge problem


Euler (1736)
Is it possible to find a cycle that includes all
the edges and vertices of the graph?
Definitions
An Euler cycle is a cycle that includes all
the edges and vertices of the graph
Theorems 8.2.17 & 8.2.18:
G has an Euler cycle if and only if G is connected
and every vertex has even degree.
Theorems 8.2.17 & 8.2.18:
G has an Euler cycle if and only if G is connected
and every vertex has even degree.
Example 4(a)
Determine if the graph has an Euler cycle.
e
b
a
c
f
d
w
g
u
v
h
 (a)   (b)   (c)   (d )   (e)   ( f )   ( g )   (h)  2
 (u )   (v)   ( w)  4
Example 4(b)
Find an Euler cycle.
e
b
a
c
f
d
w
g
u
v
h
Observation
The sum of the degrees of all the vertices
is even.
e
b
a
c
f
d
w
g
u
v
h
 (a)   (b)   (c)   (d )   (e)   ( f )   ( g )  2
 (u )   (v)   ( w)  4
Example 5 (a)
What is the sum of the degrees of all the
vertices?
v1
v2
v3
v5
6
  (v ) 
v4
i 1
v6
i
Example 5 (b)
What is the number of edges?
v1
v2
v3
v5
6
  (v ) 
v4
i 1
v6
E
i
Example 5 (c)
What is the relationship and why?
v1
v2
v3
v5
6
  (v ) 
v4
i 1
v6
E
i
Theorem 8.2.21
n
  (v )  2
i 1
i
Example 6
Is it possible to draw a graph with 6
vertices and degrees 1,1,2,2,2,3?
Corollary 8.2.22
Theorem 8.2.23
Theorem 8.2.24
b
a
v
v
u
c
u
w
x
w
x