[PDF] Euler Paths and Euler Circuits





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Euler Paths and Euler Circuits

An Euler circuit is a circuit that uses every edge of a graph exactly once. ▷ An Euler path starts and ends at different vertices. ▷ An Euler circuit starts 



Euler Paths and Euler Circuits Euler Path Euler Circuit # Odd

Euler's Theorem: 1. If a graph has more than 2 vertices of odd degree then it has no Euler paths. 2. If a 



HOW TO FIND AN EULER CIRCUIT. The book gives a proof that if a

The book gives a proof that if a graph is connected and if every vertex has even degree



Finding euler circuits in logarithmic parallel time

Cm} is an 'Eu- ler partition' of. G if each edge appears just once in its circuit see Figure 2-a. Different circuits in P may share common vertices. An. Euler.



1. For which values of n do the following graphs have an Euler

(a) Kn (b) Cn (c) Wn (d) Qn. A connected multigraph (or graph) has an Euler circuit iff each of its vertices has even degree. (a) Every vertex in Kn has degree 



Math 355 Homework 8 Key 4.5 #4 4.5 #5 4.5 #6 4.5 #7

And Euler circuit? Explain. A graph has an Euler path if at most 2 vertices have an odd degree. Since for a graph Kmn



MA115A Dr. Katiraie Section 7.1 Worksheet Name: 1. A circuit in a

The. of a vertex is the number of edges that touch that vertex. 4. According to Euler's theorem a connected graph has an Euler circuit precisely when every 



Lecture 24 Euler and Hamilton Paths Definition 1. An Euler circuit in

An Euler path in G is a simple path containing every edge of G. Definition 2. A simple path in a graph G that passes through every vertex exactly once is called 



Euler Trails and Euler Circuits

Euler Circuits. Definition. An Euler circuit is a closed Euler trail. 1. 2. 3. 4. 5. 6 a b c d e f g. 5 / 18. Page 6. Eulerian Graphs. Definition. A graph is 



Graph Theory

(a) Does G have an Euler circuit (that is an Eulerian trail)? If so



Euler Paths and Euler Circuits

An Euler circuit is a circuit that uses every edge of a graph exactly once. ? An Euler path starts and ends at different vertices.



Euler Paths and Euler Circuits

An Euler circuit is a circuit that uses every edge of a graph exactly once. ? An Euler path starts and ends at different vertices.



Euler Paths and Euler Circuits Euler Path Euler Circuit # Odd

Euler's Theorem: 1. If a graph has more than 2 vertices of odd degree then it has no Euler paths. 2. If a 



Résolution numérique une équation différentielle ordinaire ou tout

ordinaire ou tout sur la méthode d'Euler. Cas du circuit RC. Compétences visées : - Comprendre et implémenter la méthode d'Euler.



RC Euler

Résolution numérique d'une équation différentielle par la méthode d'Euler. Objectif: Saisir les paramètres du circuit : R C.



Résolution numérique déquations différentielles

Considérons par exemple le circuit électrique suivant (où le générateur est un générateur schémas d'intégration est le schéma dit d'Euler explicite.



Algorithme dEuler pour trouver un circuit hamiltonien dans le cas du

Il y a donc aussi beaucoup de circuits hamiltoniens différents. Par exemple à partir du circuit du paragraphe 6 (à gauche ci- dessous) Euler trouve un nouveau 



Résolution numérique des équations différentielles

nombreux cas la méthode d'Euler procure des résultats acceptables (au moins qualitativement). On peut en observer un exemple avec la réponse du circuit RC à 



Lecture 24 Euler and Hamilton Paths Definition 1. An Euler circuit in

An Euler path in G is a simple path containing every edge of G. Definition 2. A simple path in a graph G that passes through every vertex exactly once is called 



Euler circuit and path worksheet: Part 1: For each of these vertex

Euler circuit and path worksheet: Part 1: For each of these vertex-edge graphs try to trace it (without lifting your pen from the.



Euler Paths and Euler Circuits - University of Kansas

An Euler circuit is a circuit that uses every edge of a graph exactly once IAn Euler path starts and ends atdi erentvertices IAn Euler circuit starts and ends atthe samevertex Euler Paths and Euler Circuits B C E D A B C E D A An Euler path: BBADCDEBC Euler Paths and Euler Circuits B C E D A B C E D A Another Euler path: CDCBBADEB



Euler Path Euler Circuit - MWSU Intranet

Euler Paths and Euler Circuits Finding an Euler Circuit: There are two different ways to find an Euler circuit 1 Fleury’s Algorithm: Erasing edges in a graph with no odd vertices and keeping track of your progress to find an Euler Circuit a Begin at any vertex since they are all even A graph may have more than 1 circuit) b



Math in Society - OpenTextBookStore

DIGRAPHS AND EULER CIRCUITS 3 De nition 1 A walk in a digraph D is a sequence of arrows with the property that each arrow has source equal to the target of the previous arrow The terms cycle and so forth have the same meaning as before with the only change that we need to keep track of direction De nition 2



HOW TO FIND AN EULER CIRCUIT - University of New Mexico

vertex has even degree then there is an Euler circuit in the graph Buried in that proof is a description of an algorithm for nding such a circuit (a) First pick a vertex to the the start vertex " (b) Find at random a cycle that begins and ends at the start vertex Mark all edges on this cycle This is now your curent circuit "



Lecture 24 Euler and Hamilton Paths - Duke University

An Euler circuit has been constructed if all the edges have been used Otherwise consider the subgraph H obtained from G be deleting the edges already used and vertices that are not incident with any remaining edges



Searches related to euler circuit filetype:pdf

• An Euler circuit in a graph G is a circuit containing every edge of G once and only once › circuit - starts and ends at the same vertex • An Euler path is a path that contains every edge of G once and only once › may or may not be a circuit 3-June-02 CSE 373 - Data Structures - 24 - Paths and Circuits 9 An Euler Circuit

How do you find the path of a Euler circuit?

    The path is shown in arrows to the right, with the order of edges numbered. Euler Circuit An Euler circuit is a circuit that uses every edge in a graph with no repeats. Being a circuit, it must start and end at the same vertex. Example 6 The graph below has several possible Euler circuits.

Can Euler circuit contain repeated vertex?

    The Euler circuit can contain the repeated vertex. If we begin our path from vertex A and then go to vertices C, D or C, E, then in this process, the condition of same start and end vertex is not satisfied, but another condition of covering all edges is not satisfied.

What is a closed Euler circuit?

    If an Euler trail contains the same vertex at the start and end of the trail, then that type of trail will be known as the Euler Circuit. A closed Euler trail will be known as the Euler Circuit. Note: If all the vertices of the graph contain the even degree, then that type of graph will be known as the Euler circuit.

Is a connected multigraph an Euler circuit?

    Theorem 1. A connected multigraph with at least two vertices has an Euler circuit if and only if each of its vertices has even degree. A connected multigraph has an Euler path but not an Euler circuit if and only if it has exactly two vertices of odd degree Proof. Necessary condition for the Euler circuit.

Euler Paths and Euler Circuits

AnEuler pathis a path that uses every edge of a graph exactly once. AnEuler circuitis a circuit that uses every edge of a graph exactly once. I

An Euler path starts and ends at

dierent vertices. I

An Euler circuit starts and ends at

the same vertex.

Euler Paths and Euler CircuitsB

C D EA B C D EA

An Euler path: BBADCDEBC

Euler Paths and Euler CircuitsB

C D EA B C D EA

Another Euler path: CDCBBADEB

Euler Paths and Euler CircuitsB

C D EA B C D EA

An Euler circuit: CDCBBADEBC

Euler Paths and Euler CircuitsB

C D EA B C D EA

Another Euler circuit: CDEBBADC

Euler Paths and Euler Circuits

Is it possible to determine whether a graph has an Euler path or an Euler circuit, without necessarily having to nd one explicitly?

The Criterion for Euler Paths

Suppose that a graph has an Euler pathP.For every vertexvother than the starting and ending vertices,

the pathPentersvthesame numb erof times that it leaves v (saystimes).

Therefore, there are 2sedges havingvas an endpoint.Therefore, all vertices other than the two endpoints of

P must be even vertices.

The Criterion for Euler Paths

Suppose that a graph has an Euler pathP.For every vertexvother than the starting and ending vertices,

the pathPentersvthesame numb erof times that it leaves v (saystimes).

Therefore, there are 2sedges havingvas an endpoint.Therefore, all vertices other than the two endpoints of

P must be even vertices.

The Criterion for Euler Paths

Suppose that a graph has an Euler pathP.For every vertexvother than the starting and ending vertices,

the pathPentersvthesame numb erof times that it leaves v (saystimes).

Therefore, there are 2sedges havingvas an endpoint.Therefore, all vertices other than the two endpoints of

P must be even vertices.

The Criterion for Euler Paths

Suppose that a graph has an Euler pathP.For every vertexvother than the starting and ending vertices,

the pathPentersvthesame numb erof times that it leaves v (saystimes).

Therefore, there are 2sedges havingvas an endpoint.Therefore, all vertices other than the two endpoints of

P must be even vertices.

The Criterion for Euler Paths

Suppose the Euler pathPstarts at vertexxand ends aty.ThenPleavesxone more time than it enters, and leavesy

one fewer time than it enters.Therefore,the two endpoints of P must be odd vertices.

The Criterion for Euler Paths

Suppose the Euler pathPstarts at vertexxand ends aty.ThenPleavesxone more time than it enters, and leavesy

one fewer time than it enters.Therefore,the two endpoints of P must be odd vertices.

The Criterion for Euler Paths

Suppose the Euler pathPstarts at vertexxand ends aty.ThenPleavesxone more time than it enters, and leavesy

one fewer time than it enters.Therefore,the two endpoints of P must be odd vertices.

The Criterion for Euler Paths

The inescapable conclusion (\based on reason alone!"):

If a graphGhas an Euler path, then it must have

exactly two odd vertices.Or, to put it another way, If the number of odd vertices inGis anything other than 2, thenGcannot have an Euler path.

The Criterion for Euler Circuits

I

Suppose that a graphGhas an Euler circuitC.I

For every vertexvinG, each edge havingvas an

endpoint shows upexactly onceinC.I The circuitCentersvthe same number of times that it leavesv(saystimes), sovhas degree 2s.I

That is,v must be an even vertex.

The Criterion for Euler Circuits

I

Suppose that a graphGhas an Euler circuitC.I

For every vertexvinG, each edge havingvas an

endpoint shows upexactly onceinC.I The circuitCentersvthe same number of times that it leavesv(saystimes), sovhas degree 2s.I

That is,v must be an even vertex.

The Criterion for Euler Circuits

I

Suppose that a graphGhas an Euler circuitC.I

For every vertexvinG, each edge havingvas an

endpoint shows upexactly onceinC.I The circuitCentersvthe same number of times that it leavesv(saystimes), sovhas degree 2s.I

That is,v must be an even vertex.

The Criterion for Euler Circuits

I

Suppose that a graphGhas an Euler circuitC.I

For every vertexvinG, each edge havingvas an

endpoint shows upexactly onceinC.I The circuitCentersvthe same number of times that it leavesv(saystimes), sovhas degree 2s.I

That is,v must be an even vertex.

The Criterion for Euler Circuits

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 inGis anything other than 0, thenGcannot have an Euler circuit.

Things You Should Be Wondering

I

Doeseverygraph withzeroodd vertices have an Euler

circuit? I

Doeseverygraph withtwoodd vertices have an Euler

path? I Is it possible for a graph have justoneodd vertex?

How Many Odd Vertices?

How Many Odd Vertices?Odd vertices

How Many Odd Vertices?86

2 4 86

Number of odd vertices

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 arenverticesV1;:::;Vn, with degreesd1;:::;dn, and there areeedges, then d

1+d2++dn1+dn= 2e

Or, equivalently,

e=d1+d2++dn1+dn2

The Handshaking Theorem

Why \Handshaking"?

Ifnpeople shake hands, and theithperson shakes handsdi times, then the total number of handshakes that take place is d

1+d2++dn1+dn2

(How come? Each handshake involves two people, so the numberd1+d2++dn1+dncounts every handshake twice.)

The Number of Odd Vertices

I

The number of edges in a graph is

d

1+d2++dn2

which must be aninteger.I

Therefore,d1+d2++dnmust be aneven number.I

Therefore, the numbersd1;d2;;dnmust include an

even number of odd numbers.I

Every graph has an even number of odd vertices!

The Number of Odd Vertices

I

The number of edges in a graph is

d

1+d2++dn2

which must be aninteger.I

Therefore,d1+d2++dnmust be aneven number.I

Therefore, the numbersd1;d2;;dnmust include an

even number of odd numbers.I

Every graph has an even number of odd vertices!

The Number of Odd Vertices

I

The number of edges in a graph is

d

1+d2++dn2

which must be aninteger.I

Therefore,d1+d2++dnmust be aneven number.I

Therefore, the numbersd1;d2;;dnmust include an

even number of odd numbers.I

Every graph has an even number of odd vertices!

The Number of Odd Vertices

I

The number of edges in a graph is

d

1+d2++dn2

which must be aninteger.I

Therefore,d1+d2++dnmust be aneven number.I

Therefore, the numbersd1;d2;;dnmust include an

even number of odd numbers.I

Every graph has an even number of odd vertices!

Back to Euler Paths and Circuits

Here's what we know so far:# odd verticesEuler path?Euler circuit?

0NoMaybe

2MaybeNo

4, 6, 8, ...NoNo

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

Can we give a better answer than \maybe"?

Euler Paths and Circuits | The Last Word

Here is the answer Euler gave:# odd verticesEuler path?Euler circuit?

0NoYes*2Yes*No

4, 6, 8, ...NoNo

1, 3, 5,No such graphs exist

*Provided the graph is connected.Next question: If an Euler path or circuit exists, how do you nd it?

Euler Paths and Circuits | The Last Word

Here is the answer Euler gave:# odd verticesEuler path?Euler circuit?

0NoYes*2Yes*No

4, 6, 8, ...NoNo

1, 3, 5,No such graphs exist

*Provided the graph is connected.Next question: If an Euler path or circuit exists, how do you nd it?

Bridges

Removing a single edge from a connected graph can make it disconnected. Such an edge is called abridge.

Bridges

Removing a single edge from a connected graph can make it disconnected. Such an edge is called abridge.

Bridges

Removing a single edge from a connected graph can make it disconnected. Such an edge is called abridge.

Bridges

Loops cannot be bridges, because removing a loop from a graph cannot make it disconnected.delete loop e e

Bridges

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

Bmultiple

delete

Bridges

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

Bmultiple

delete

Finding Euler Circuits and Paths

\Don't burn your bridges."

Finding Euler Circuits and Paths

Problem: Find an Euler circuit in the graph below.AB F ED C

Finding Euler Circuits and Paths

There are two odd vertices, A and F. Let's start at F.AB F ED C

Finding Euler Circuits and Paths

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

Finding Euler Circuits and Paths

Path so far: FEAB

F ED C

Finding Euler Circuits and Paths

Path so far: FEAAB

F ED C

Finding Euler Circuits and Paths

Path so far: FEACAB

F ED C

Finding Euler Circuits and Paths

Path so far: FEACBAB

quotesdbs_dbs17.pdfusesText_23
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