The Konigsberg Network

Do you know that geometry has been applied to problems in designing telephone and communication system, transportation connection and computer connection? these design problems can often be represented by a network consists of a set of vertices and the connecting paths between them

One of the most famous network problems was noted in the 16th century in konigsberg, Prussia(Prussia covered the teritory under the former german empire). The town comprised two island in the middle of a river. The town was connected by 7 bridges which crossed the river. The question was whether person could plan a walk, starting any where, so that all 7 bridges could be crossed without crossing the same bridge twice.

Leonhard Euler first studied this problem in 1735. Here is Euler's approach to the problem: Euler began by redrawing the map so that 4 areas of land were represented by points with seven lines to represent the bridges. That konigsberg problem can now restated as follows: Given 4 distinct vertices. try to draw a continuous path from one vertex to another, without retracing a previous drawn path.

Euler discovered that such a network can only be drawn if one of these conditions are true:

1. there are exactly two vertices with an odd number of paths leading from them.

2. All vertices have an even no. of paths leading of them.