MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles
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Transcript of MAT 2720 Discrete Mathematics Section 8.2 Paths and Cycles
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MAT 2720Discrete Mathematics
Section 8.2Paths and Cycles
http://myhome.spu.edu/lauw
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Goals Paths and Cycles
•Definitions and Examples•More Definitions
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Definitions
0vnv
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1nv
2v
1v
Definitions
0vnv
3v1e
3e2e
ne
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1nv
2v
1v
Definitions
0vnv
3v
0 1 2, , , , nv v v v
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Example 1(a) Write down a path from b to e with
length 4.
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Example 1(b) Write down a path from b to e with
length 5.
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Example 1(c) Write down a path from b to e with
length 6.
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Definitions
vw
vw
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Example 2The graph is not connected because …
a
bc d
e f
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Definitions
ev
w
v
we
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Definitions
ev
w
ev
v
we
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Definitions
ev
w
( ) , is a graph.b V E
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Example 3How many subgraphs are there with 3 edges?
a
bc
e f
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Definitions
v
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Definitions
v
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Connected Graph & Component
v
What can we say about the components of a graph if it is connected?
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Connected Graph & Component
v
What can we say about the graph if it has exactly one component?
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Theorem
v
A graph is connected if and only if it has exactly one component
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Definitions
vw
u
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Definitions
v
wu
x
v
wu
x
ab
c
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Definitions
v
wu
x
v
wu
x
ab
c
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DefinitionsThe degree of a vertex v, denoted by (v), is the number of edges incident on v
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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
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The Königsberg bridge problem Euler (1736) Is it possible to cross all seven bridges just once
and return to the starting point?
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The Königsberg bridge problem Edges represent bridges and each
vertex represents a region.
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The Königsberg bridge problem Euler (1736) Is it possible to find a cycle that includes
all the edges and vertices of the graph?
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DefinitionsAn Euler cycle is a cycle that includes all the edges and vertices of the graph
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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.
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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.
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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
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Example 4(b)
v
w
ua
b c d
e f
g
h
Find an Euler cycle.
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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.
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Example 5 (a)What is the sum of the degrees of all the vertices?
6
1
( )ii
v
1v
2v3v
4v
5v 6v
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Example 5 (b)What is the number of edges?
1v
2v
E
3v4v
5v 6v
6
1
( )ii
v
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Example 5 (c)What is the relationship and why?
1v
2v
E
3v4v
5v 6v
6
1
( )ii
v
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Theorem 8.2.21
1
( ) 2n
ii
v
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Example 6Is it possible to draw a graph with 6 vertices and degrees 1,1,2,2,2,3?
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Corollary 8.2.22
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Theorem 8.2.23
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Theorem 8.2.24
v
wu
x
v
wu
x
ab
c