Euler’s Theorems and Fleury’s Algorithm -...

Euler’s Theorems and Fleury’s AlgorithmLecture 27Section 5.3

Robb T. Koether

Hampden-Sydney College

Fri, Oct 27, 2017

Robb T. Koether (Hampden-Sydney College) Euler’s Theorems and Fleury’s Algorithm Fri, Oct 27, 2017 1 / 19

1 Definitions

2 Euler’s Theorems

3 Fleury’s Algorithm

4 The Splicing Algorithm

5 The Mail Carrier Problem Solved

6 Assignment

Outline

1 Definitions

6 Assignment

Euler Paths and Circuits

Definition (Euler Path)An Euler path (pronounced "oiler") is a path that traverses each edgeexactly once.

Definition (Euler Circuit)An Euler circuit is an Euler path that is a circuit.

Euler Paths and Circuits

In the Bridges of Königsberg Problem, we seek an Euler path andan Euler circuit.

Definitions

Definition (Connected)A graph is connected if, for any two vertices, there is a path from oneto the other.

Definition (Even and Odd Vertices)A vertex is even if an even number of edges emanate from it.Otherwise, it is odd.

Outline

1 Definitions

6 Assignment

Euler’s Theorems

Theorem (Euler Circuits)If a graph is connected and every vertex is even, then it has an Eulercircuit. Otherwise, it does not have an Euler circuit.

Euler’s Theorems

Theorem (Euler Paths)If a graph is connected and it has exactly 2 odd vertices, then it has anEuler path and any Euler path must begin at one of the odd verticesand end that the other one.

Euler’s Theorems

Theorem (Euler Paths)If a graph is not connected or if it has more than 2 odd vertices, then itdoes not have an Euler path.

Outline

1 Definitions

6 Assignment

Fleury’s AlgorithmBegin at any vertex and follow an edge.Whenever there is a choice of edges, choose an edge that doesnot separate you from any remaining (unvisited) edges.

(Thisrequires thought.)Continue to follow edges in this manner.When you reach the starting point, you have an Euler circuit.

Fleury’s AlgorithmBegin at any vertex and follow an edge.Whenever there is a choice of edges, choose an edge that doesnot separate you from any remaining (unvisited) edges. (Thisrequires thought.)

Continue to follow edges in this manner.When you reach the starting point, you have an Euler circuit.

Fleury’s AlgorithmBegin at any vertex and follow an edge.Whenever there is a choice of edges, choose an edge that doesnot separate you from any remaining (unvisited) edges. (Thisrequires thought.)Continue to follow edges in this manner.When you reach the starting point, you have an Euler circuit.

Fleury’s Algorithm

Outline

1 Definitions

6 Assignment

The Splicing AlgorithmBegin at any vertex and follow an edge.

Continue to follow edges in any way whatsoever until you reachthe starting point, at which point you have a circuit.

(This does notrequire thought.)

If there are untraveled edges, then retrace the circuit until youreach the first untraveled edge.Follow untraveled edges, as before, until you return to that vertex.“Splice” that circuit into the original circuit.Repeat the last three steps until there are no more untravelededges. You now have an Euler circuit.

The Splicing AlgorithmBegin at any vertex and follow an edge.Continue to follow edges in any way whatsoever until you reachthe starting point, at which point you have a circuit.

(This does notrequire thought.)If there are untraveled edges, then retrace the circuit until youreach the first untraveled edge.Follow untraveled edges, as before, until you return to that vertex.“Splice” that circuit into the original circuit.Repeat the last three steps until there are no more untravelededges. You now have an Euler circuit.

The Splicing AlgorithmBegin at any vertex and follow an edge.Continue to follow edges in any way whatsoever until you reachthe starting point, at which point you have a circuit. (This does notrequire thought.)

If there are untraveled edges, then retrace the circuit until youreach the first untraveled edge.Follow untraveled edges, as before, until you return to that vertex.“Splice” that circuit into the original circuit.Repeat the last three steps until there are no more untravelededges. You now have an Euler circuit.

The Splicing AlgorithmBegin at any vertex and follow an edge.Continue to follow edges in any way whatsoever until you reachthe starting point, at which point you have a circuit. (This does notrequire thought.)If there are untraveled edges, then retrace the circuit until youreach the first untraveled edge.

Follow untraveled edges, as before, until you return to that vertex.“Splice” that circuit into the original circuit.Repeat the last three steps until there are no more untravelededges. You now have an Euler circuit.

The Splicing AlgorithmBegin at any vertex and follow an edge.Continue to follow edges in any way whatsoever until you reachthe starting point, at which point you have a circuit. (This does notrequire thought.)If there are untraveled edges, then retrace the circuit until youreach the first untraveled edge.Follow untraveled edges, as before, until you return to that vertex.

“Splice” that circuit into the original circuit.Repeat the last three steps until there are no more untravelededges. You now have an Euler circuit.

The Splicing AlgorithmBegin at any vertex and follow an edge.Continue to follow edges in any way whatsoever until you reachthe starting point, at which point you have a circuit. (This does notrequire thought.)If there are untraveled edges, then retrace the circuit until youreach the first untraveled edge.Follow untraveled edges, as before, until you return to that vertex.“Splice” that circuit into the original circuit.

Repeat the last three steps until there are no more untravelededges. You now have an Euler circuit.

The Splicing AlgorithmBegin at any vertex and follow an edge.Continue to follow edges in any way whatsoever until you reachthe starting point, at which point you have a circuit. (This does notrequire thought.)If there are untraveled edges, then retrace the circuit until youreach the first untraveled edge.Follow untraveled edges, as before, until you return to that vertex.“Splice” that circuit into the original circuit.Repeat the last three steps until there are no more untravelededges. You now have an Euler circuit.

The Splicing Algorithm

Outline

1 Definitions

6 Assignment

The Mail Carrier Problem Solved

The mail carrier’s graph never contains any odd vertices.

4 4 4 4 4 22

2 4 4 2

Outline

1 Definitions

6 Assignment

Assignment

AssignmentChapter 5: Exercises 29, 30, 31, 35, 36, 37, 38, 39.

Euler’s Theorems and Fleury’s Algorithm -...

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