The Konigsberg Problem Introduction to Graph Theory

Ali Mahmudi, Introduction to Graph Theory [1]

Chapter 1 Introduction to Graph Theory

D r . Al i Ma h m u d i Ju r u sa n Pen d i d i k a n M a t em a t i k a FM I PA U NY To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be. Leonhard Euler 1707-1783

A. The Konigsberg Problem

Konigsberg is a city which was the capital of East Prussia but now is known as Kaliningrad in Russia. The city is built around the River Pregel where it joins another river. An island named Kniephof is in the middle of where the two rivers join. There are seven bridges that join the different parts of the city on both sides of the rivers and the island as shown in the Figure 1.1. Sumber: http:math.youngzones.orgKonigsberg.html Figure 1.1. Konigsberg Bridge Ali Mahmudi, Introduction to Graph Theory [2] People tried to find a way to walk all seven bridges without crossing a bridge twice. This problem was called Konigsberg bridge problem: is there a walking route that crosses each of the seven bridges of Konigsberg exactly once? The problem came to the attention of a Swiss mathematician named Leonhard Euler pronounced oiler. In 1735, Euler presented the solution to the problem. He explained why crossing all seven bridges without crossing a bridge twice was impossible. While solving this problem, he developed a new mathematics field called graph theory, which later served as the basis for another mathematical field called topology. The key to Euler’s solution was very simple abstraction of the problem. Euler redraw the diagram of the city of Konigsberg by representing each of the land masses as a vertex and representing each bridge as an edge connecting the vertices corresponding to the land masses as shown in the Figure 1.2. By this method we have a graph that encodes the necessary information. The problem reduces to finding a “closed walk” in the graph which traverses each edge exactly once. This is called an Eulerian circuit. Figure 1.2. Graph as a Model of Konigsberg Problem Leonhard Euler 1707-1783 Sumber: http:www.pdmi.ras.ru Ali Mahmudi, Introduction to Graph Theory [3] The field of graph theory began to blossom in the twentieth century as more and more modeling possibilities we recognized – and growth continues. In the mid 1800s, people began to realize that graphs could be used to model many things that were of interest in society. For example, the “Four Color Maps Conjecture”, introduced by DeMorgan in 1852, was a famous problem that was seemingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently. This conjecture can easily be phrased in terms of graph theory, and many researchers used this approach during the dozen decades that the problem remained unsolved.

B. The Definition of Graph