Eular Path

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Euler Paths and Euler Circuits

An Euler path is a path that uses every edge of a graphexactly once.

An Euler circuit is a circuit that uses every edge of a graphexactly once.

An Euler path starts and ends at differentvertices. An Euler circuit starts and ends atthe samevertex.


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B

C

DE

A

B

C

DE

A

An Euler path: BBADCDEBC


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B

C

DE

A

B

C

DE

A

Another Euler path: CDCBBADEB


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B

C

DE

A

B

C

DE

A

An Euler circuit: CDCBBADEBC


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B

C

DE

A

B

C

DE

A

Another Euler circuit: CDEBBADC


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Is it possible to determine whether a graph has an

Euler path or an Euler circuit, without necessarilyhaving to find one explicitly?


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The Criterion for Euler Paths

Suppose that a graph has an Euler path P.


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For every vertex vother than the starting and ending vertices,the path P enters v thesamenumber of times that it leaves v

(say s times).


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(say s times).

Therefore, there are 2s edges having vas an endpoint.


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(say s times).

Therefore, there are 2s edges having vas an endpoint.

Therefore, all vertices other than the two endpoints ofP must be even vertices.


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Suppose the Euler path Pstarts at vertex xand ends at y.


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Then P leaves xone more time than it enters, and leaves yone fewer time than it enters.


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Then P leaves xone more time than it enters, and leaves yone fewer time than it enters.

Therefore, the two endpoints of P must be odd vertices.


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The inescapable conclusion (based on reason alone!):

If a graph G has an Euler path, then it must have

exactly two odd vertices.

Or, to put it another way,

If the number of odd vertices in G is anything other

than 2, then Gcannot have an Euler path.


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The Criterion for Euler Circuits

Suppose that a graph Ghas an Euler circuit C.


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For every vertex v in G, each edge having vas an

endpoint shows up exactly once in C.


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The circuit C enters vthe same number of times that itleaves v (say s times), so vhas degree 2s.


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The circuit C enters vthe same number of times that itleaves v (say s times), so vhas degree 2s.

That is, v must be an even vertex.


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The inescapable conclusion (based on reason alone):

If a graphGhas an Euler circuit, then all of its vertices

must be even vertices.

Or, to put it another way,

If the number of odd vertices in G is anything other

than 0, then G cannot have an Euler circuit.


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Things You Should Be Wondering

Does every graph with zero odd vertices have an Eulercircuit?

Does every graph with two odd vertices have an Eulerpath?

Is it possible for a graph have just one odd vertex?


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How Many Odd Vertices?


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Odd vertices


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86

2 4 8

6Number of odd vertices


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The Handshaking Theorem

The Handshaking Theorem says that

In every graph, the sum of the degrees of all vertices

equals twice the number of edges.

If there are n vertices V1, . . . , Vn, with degrees d1, . . . , dn, andthere are eedges, then

d1+d2+ +dn1+dn = 2e

Or, equivalently,

e= d1+d2+ +dn1+dn

2


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The Handshaking Theorem

Why Handshaking?

Ifn people shake hands, and the ith person shakes hands ditimes, then the total number of handshakes that take place is

d1+d2+ +dn1+dn2

.

(How come? Each handshake involves two people, so thenumber d1+d2+ +dn1+dn counts every handshaketwice.)


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The Number of Odd Vertices

The number of edges in a graph is

d1+d2+ +dn2

which must be an integer.


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d1+d2+ +dn2


Therefore, d1+d2+ +dn must be an even number.


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d1+d2+ +dn2



Therefore, the numbers d1, d2, , dn must include aneven number of odd numbers.


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d1+d2+ +dn2



Therefore, the numbers d1, d2, , dn must include aneven number of odd numbers.

Every graph has an even number of odd vertices!


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Back to Euler Paths and Circuits

Heres what we know so far:

# odd vertices Euler path? Euler circuit?

0 No Maybe2 Maybe No

4, 6, 8, . . . No No

1, 3, 5, . . . No such graphs exist!

Can we give a better answer than maybe?


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Euler Paths and Circuits The Last Word

Here is the answer Euler gave:


0 No Yes*

2 Yes* No4, 6, 8, . . . No No

1, 3, 5, No such graphs exist

* Provided the graph is connected.


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Euler Paths and Circuits The Last Word

Here is the answer Euler gave:


0 No Yes*

2 Yes* No4, 6, 8, . . . No No

1, 3, 5, No such graphs exist

* Provided the graph is connected.

Next question: If an Euler path or circuit exists, how do

you find it?

B d


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Bridges

Removing a single edge from a connected graph can make it

disconnected. Such an edge is called a bridge.

B id


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Bridges



B id


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Bridges



B id


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Bridges

Loops cannot be bridges, because removing a loop from agraph cannot make it disconnected.

deleteloop e

e

B id


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Bridges

If two or more edges share both endpoints, then removing anyone of them cannot make the graph disconnected. Therefore,none of those edges is a bridge.

edgesA

C

D

B

multipledelete

B id


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Bridges

If two or more edges share both endpoints, then removing anyone of them cannot make the graph disconnected. Therefore,none of those edges is a bridge.

edgesA

C

D

B

multipledelete

Finding E ler Circ its and Paths


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Finding Euler Circuits and Paths

Dont burn your bridges.



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Problem: Find an Euler circuit in the graph below.

A

B

FE

DC



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There are two odd vertices, A and F. Lets start at F.

A

B

FE

DC



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Start walking at F. When you use an edge, delete it.

A

B

FE

DC



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Path so far: FE

A

B

FE

DC



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Path so far: FEA

A

B

FE

DC



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Path so far: FEAC

A

B

FE

DC



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Path so far: FEACB

A

B

FE

DC



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Up until this point, the choices didnt matter.



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But now, crossing the edge BA would be a mistake, becausewe would be stuck there.



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But now, crossing the edge BA would be a mistake, becausewe would be stuck there.

The reason is that BA is a bridge. We dont want to cross(burn?) a bridge unless it is the only edge available.



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g u u s s

Path so far: FEACB

A

B

FE

DC



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g

Path so far: FEACBD.

A

B

FE

DC



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g

Path so far: FEACBD. Dont cross the bridge!

A

B

FE

DC



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g

Path so far: FEACBDC

A

B

FE

DC



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g

Path so far: FEACBDC Now we have to cross the bridge CF.

A

B

FE

DC



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Path so far: FEACBDCF

A

B

FE

DC



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Path so far: FEACBDCFD

A

B

FE

DC



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Path so far: FEACBDCFDB

A

B

FE

DC



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Euler Path: FEACBDCFDBA

A

B

FE

DC



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Euler Path: FEACBDCFDBA

A

B

FE

DC

Fleurys Algorithm


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To find an Euler path or an Euler circuit:

Fleurys Algorithm


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1. Make sure the graph has either 0 or 2 odd vertices.

Fleurys Algorithm


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2. If there are 0 odd vertices, start anywhere. If there are 2

odd vertices, start at one of them.

Fleurys Algorithm


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3. Follow edges one at a time. If you have a choice betweena bridge and a non-bridge,always choose the non-bridge.

Fleurys Algorithm


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4. Stop when you run out of edges.

Fleurys Algorithm


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4. Stop when you run out of edges.

This is called Fleurys algorithm, and it always works!

Fleurys Algorithm: Another Example


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K LJ M

HE

A

F

CB D

NO P

Q

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