The Seven Bridges

In the summer of 1735, Swiss mathematician Leonard Euler walks across one of the many bridges in the city of Konigsberg. Being situated on the river Pregel, Konigsberg has seven bridges connecting two island in the river with the riverbanks north and south. After, brief walk and some frustration, Euler was curious to see is it possible to travel or tour all seven bridges but only crossing each bridge once.

Konigsberg
A graph that represents the bridges of Konigsberg.

Euler starts by simplifying the problem. He begins by representing each land mass a circle and each bridge as a line. The circles he calls nodes and lines he calls edges. The whole diagram is called a graph. By doing so he removes all unneeded information and focuses only on how each bridge relates with each landmass.

Euler soon realizes that for such a tour to be possible all nodes with the exception of at most two must have an even number of edges. If a person traverses an edge to reach a node they must cross a different edge to leave, if there is an odd then one edge will have to be traversed twice. Since all four nodes had odd numbers of edges no such tour existed.

Konigsberg_bridges
Konigsberg, now Kaliningrad, was home to the seven bridges one which inspired Euler’s first paper on graph theory published in 1736. Unfortunately two of the five bridges were destroyed in WW2 

Though unable to find a successful tour, Euler inadvertently creates the field of Graph theory. This field would go on to provide the mathematical structure and frame work for fields such as electrical engineering, computer science and even social networking. However given any graph, a path that traverses all the edges once and only once is called an Euler Tour.

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