Euler Paths and Euler Circuits - University of Kansas

The Number of Odd Verti

The number of edges in a
d1 + d2

which must be an integer.

ices

graph is
2 + · · · + dn

2
.

The Number of Odd Verti

The number of edges in a
d1 + d2

which must be an integer.
Therefore, d1 + d2 + · · · +

ices

graph is
2 + · · · + dn

2
.
+ dn must be an even number.

The Number of Odd Verti

The number of edges in a
d1 + d2

which must be an integer.
Therefore, d1 + d2 + · · · +
Therefore, the numbers d1,
even number of odd num

ices

graph is
2 + · · · + dn

2
.

+ dn must be an even number.
, d2, · · · , dn must include an
mbers.

The Number of Odd Verti

The number of edges in a
d1 + d2

which must be an integer.
Therefore, d1 + d2 + · · · +
Therefore, the numbers d1,
even number of odd num
Every graph has an even

ices

graph is
2 + · · · + dn

2
.

+ dn must be an even number.
, d2, · · · , dn must include an
mbers.

n number of odd vertices!

Back to Euler Paths and C

Here’s what we know so far:

# odd vertices Euler

0 N
2 Ma
4, 6, 8, . . .
1, 3, 5, . . . N
N

Can we give a better answer

Circuits

r path? Euler circuit?
No Maybe
aybe No
No No
No such graphs exist!

r than “maybe”?

Euler Paths and Circuits —

Here is the answer Euler gave:

# odd vertices Euler

0 N
2 Ye
4, 6, 8, . . . N

1, 3, 5, N

* Provided the graph is connect

— The Last Word

r path? Euler circuit?

No Yes*
Yes* No
No No

No such graphs exist

ted.

Euler Paths and Circuits —

Here is the answer Euler gave:

# odd vertices Euler

0 N
2 Ye
4, 6, 8, . . . N

1, 3, 5, N

* Provided the graph is connect

Next question: If an Euler p
you find it?

— The Last Word

r path? Euler circuit?

No Yes*
Yes* No
No No

No such graphs exist

ted.
path or circuit exists, how do

Bridges

Removing a single edge from a
disconnected. Such an edge is c

connected graph can make it
called a bridge.

Bridges

Removing a single edge from a
disconnected. Such an edge is c

connected graph can make it
called a bridge.

Bridges

Removing a single edge from a
disconnected. Such an edge is c

connected graph can make it
called a bridge.

Bridges

Loops cannot be bridges, becau
graph cannot make it disconnec

e

dele
loop

use removing a loop from a
cted.

ete
pe

Bridges

If two or more edges share both
one of them cannot make the g
none of those edges is a bridge.

C dele
A mult
edg
B
D

h endpoints, then removing any
graph disconnected. Therefore,
.

ete
tiple
ges

Bridges

If two or more edges share both
one of them cannot make the g
none of those edges is a bridge.

C dele
A mult
edg
B
D

h endpoints, then removing any
graph disconnected. Therefore,
.

ete
tiple
ges

Finding Euler Circuits and
“Don’t burn y

d Paths
your bridges.”

Finding Euler Circuits and

Problem: Find an Euler circuit i

A
C

EF

d Paths

in the graph below.

B

D
F

Finding Euler Circuits and

There are two odd vertices, A a

A
C

EF

d Paths

and F. Let’s start at F.

B

D
F

Finding Euler Circuits and

Start walking at F. When you u

A
C

EF

d Paths

use an edge, delete it.

B

D
F

Finding Euler Circuits and

Path so far: FE

A
C

EF

d Paths

B

D
F

Finding Euler Circuits and

Path so far: FEA

A
C

EF

d Paths

B

D
F

Finding Euler Circuits and

Path so far: FEAC

A
C

EF

d Paths

B

D
F

Finding Euler Circuits and

Path so far: FEACB

A
C

EF

d Paths

B

D
F

Finding Euler Circuits and

Up until this point, the choices

d Paths

didn’t matter.

Finding Euler Circuits and

Up until this point, the choices

But now, crossing the edge BA
we would be stuck there.

d Paths

didn’t matter.
would be a mistake, because

Finding Euler Circuits and

Up until this point, the choices

But now, crossing the edge BA
we would be stuck there.

The reason is that BA is a brid
(“burn”?) a bridge unless it is t

d Paths

didn’t matter.
would be a mistake, because

dge. We don’t want to cross
the only edge available.

Finding Euler Circuits and

Path so far: FEACB

A
C

EF

d Paths

B

D
F