Bridge problem : Discrete Structure

BRIDGE PROBLEM

By:-

Mitul K. Desai - 23

WHAT IS IT ?• In the early eighteenth century, the mediaeval town of

Königsberg in Prussia had a central island (the Kneiphof) around which the Pregel river flowed before dividing in two.

• The four parts of the town were linked by seven bridges as shown on the diagram.

7 bridges of Koenigsberg (1736)

HISTORY• The official birth of graph theory goes

back precisely to the year 1736, when mathematician Leonard Euler (1707-1783) was asked the following question at the court of the king of Prussia:

QUESTION ?• Is it possible for a person to take a walk around town,

starting and ending at the same location, and crossing each of the seven bridges exactly once ?

SOLUTION• Euler studied this problem using the multigraph obtained

where:

• the four regions are represented by vertices.

• the bridges by edges.

DEFINATION• An Euler circuit in a graph G is a simple circuit containing

every edge of G. An Euler path in G is a simple path containing every edge of G.

• Path: a path from v to w is a walk from v to w that does not have any repeated edges.

• Circuit: a circuit is a closed walk, with no repeated edges.

Euler’s Theorem 1

• If a graph has any odd vertices, then it cannot have an Euler circuit.

• If a graph is connected and every vertex is even, then it has at least one Euler circuit.

CONCLUSION

• What we can show is that every vertex must have even degree.

• First note that an Euler circuit begins with a vertex a and continues with an edge incident with a, say {a, b}.

• The edge {a, b} contributes one to degree(a).

CONCLUSION• Each time the circuit passes through a vertex it contributes two

to the vertex’s degree, because the circuit enters via an edge incident with this vertex and leaves via another such edge.

• Finally, the circuit terminates where it started, contributing one to degree(a).

• Therefore, degree(a) must be even, because the circuit contributes one when it begins, one when it ends, and two every time it passes through a

Thank You