Königsberg, Euler and the origins of graph theory

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Konigsberg, Euler and the origins of graph theory

Philip Puylaert

February 2014

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Konigsberg, East Prussia

capital of East Prussia (1457–1945)

Pregel river

university

birth place of Immanuel Kant, David Hilbert, Kathe Kollwitz

destroyed at the end of World War II

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Nowadays: Kaliningrad

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Nowadays: Kaliningrad

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Nowadays: Kaliningrad

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

The 7 bridges of Konigsberg

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

The 7 bridges of Konigsberg

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Leonhard Euler

Basel 1707 – St.-Petersburg 1783

professor at 20

enormously productive

influence found everywhere in mathand physics

most famous formula: 1 + e iπ = 0

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

The 7 bridges problem

A

B

C

D

Definitions

graph

vertices (singular: vertex) — edges

order of a vertex

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

The 7 bridges problem

A

B

C

D

Definitions

graph

vertices (singular: vertex) — edges

order of a vertex

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

The 7 bridges problem

A

B

C

D

Definitions

graph

vertices (singular: vertex) — edges

order of a vertex

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

When can you take the desired walk?

A

1

2

3

4

vertex of even order

A

1

23

vertex of odd order

The graph is traversable

if all vertices have even order→ Euler tour, a closed walk

if exactly 2 vertices have odd order→ use them to start and finish your walk

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

When can you take the desired walk?

A

1

2

3

4

vertex of even order

A

1

23

vertex of odd order

The graph is traversable

if all vertices have even order→ Euler tour, a closed walk

if exactly 2 vertices have odd order→ use them to start and finish your walk

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

When can you take the desired walk?

A

1

2

3

4

vertex of even order

A

1

23

vertex of odd order

The graph is traversable

if all vertices have even order→ Euler tour, a closed walk

if exactly 2 vertices have odd order→ use them to start and finish your walk

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Examples of traversable graphs

The graph is traversable

if all vertices have even order→ Euler tour, a closed walk

if exactly 2 vertices have odd order→ use them to start and finish your walk

A

BC

1

2

3

A B

CD

1

2

3

4 5

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Examples of traversable graphs

The graph is traversable

if all vertices have even order→ Euler tour, a closed walk

if exactly 2 vertices have odd order→ use them to start and finish your walk

A

BC

1

2

3

A B

CD

1

2

3

4 5

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Back to the 7 bridges problem

A

B

C

D

the order of A is 3

the order of B is 4

the order of C is 3

the order of D is 3

Conclusion

The graph of the 7 bridges problem is not traversable.It’s impossible to take a walk crossing every bridge exactly once.

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Back to the 7 bridges problem

A

B

C

D

the order of A is 3

the order of B is 4

the order of C is 3

the order of D is 3

Conclusion

The graph of the 7 bridges problem is not traversable.It’s impossible to take a walk crossing every bridge exactly once.

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Back to the 7 bridges problem

A

B

C

D

the order of A is 3

the order of B is 4

the order of C is 3

the order of D is 3

Conclusion

The graph of the 7 bridges problem is not traversable.It’s impossible to take a walk crossing every bridge exactly once.

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Application 1: traffic

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Application 2: social networks

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Application 2: social networks

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Application 3: ranking of search results by Google

each vertex represents a web pagearrow D → A means: page D contains a link to page A

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

Summary

What have you learned in this slidecast?

basic concepts of graph theory: graph, vertex, edge, order of avertex

you and Euler solved the 7 bridges problem by proving when agraph is traversablethe Konigsberg graph is not traversable

some applications of graph theory, e.g. traffic, social networks

History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading

More information?

Reinhard Diestel, Graph Theory (3rd edition), Springer Verlag,2005www.math.ubc.ca/~solymosi/2007/443/GraphTheoryIII.pdf

Fred Buckley, A Friendly Introduction to Graph Theory,Prentice Hall, 2002

Glen Gray, Graph Theory 1 — Intro via Konigsberg Bridgewww.youtube.com/watch?v=BK kYjFWWX0