MAT 2720Discrete Mathematics

Section 8.2Paths and Cycles

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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