MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles http://myhome.spu.edu/lauw
MAT 2720Discrete Mathematics
Section 8.2Paths and Cycles
http://myhome.spu.edu/lauw
Goals Paths and Cycles
•Definitions and Examples•More Definitions
Definitions
0vnv
1nv
2v
1v
Definitions
0vnv
3v1e
3e2e
ne
1nv
2v
1v
Definitions
0vnv
3v
0 1 2, , , , nv v v v
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
vw
vw
Example 2The graph is not connected because …
a
bc d
e f
Definitions
ev
w
v
we
Definitions
ev
w
ev
v
we
Definitions
ev
w
( ) , is a graph.b V E
Example 3How many subgraphs are there with 3 edges?
a
bc
e f
Definitions
v
Definitions
v
Connected Graph & Component
v
What can we say about the components of a graph if it is connected?
Connected Graph & Component
v
What can we say about the graph if it has exactly one component?
Theorem
v
A graph is connected if and only if it has exactly one component
Definitions
vw
u
Definitions
v
wu
x
v
wu
x
ab
c
Definitions
v
wu
x
v
wu
x
ab
c
DefinitionsThe degree of a vertex v, denoted by (v), is the number of edges incident on v
Definitions
v
w
u
The degree of a vertex v, denoted by (v), is the number of edges incident on v
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )a b c d e f g hu v w
a
b c d
e f
g
h
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?
DefinitionsAn 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)
v
w
ua
b c d
e f
g
h
Determine if the graph has an Euler cycle.
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2( ) ( ) ( ) 4a b c d e f g hu v w
Example 4(b)
v
w
ua
b c d
e f
g
h
Find an Euler cycle.
Observation
v
w
u
( ) ( ) ( ) ( ) ( ) ( ) ( ) 2( ) ( ) ( ) 4a b c d e f gu v w
a
b c d
e f
g
h
The sum of the degrees of all the vertices is even.
Example 5 (a)What is the sum of the degrees of all the vertices?
6
1
( )ii
v
1v
2v3v
4v
5v 6v
Example 5 (b)What is the number of edges?
1v
2v
E
3v4v
5v 6v
6
1
( )ii
v
Example 5 (c)What is the relationship and why?
1v
2v
E
3v4v
5v 6v
6
1
( )ii
v
Theorem 8.2.21
1
( ) 2n
ii
v
Example 6Is 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
v
wu
x
v
wu
x
ab
c