Real Mathematics: Graphs #1

Puzzles and mathematics have a deeper relationship than one might assume. When you take a look at history of mathematics, it is not that rare to see a puzzle causing mathematics to change.

Without a doubt, Seven Bridges of Königsberg (1736) is the most famous one of those stories.

Leonhard Euler, one of the most influential mathematicians of all times, was given this puzzle/problem in the 18^th century. Euler had a great reputation for solving “unsolvable” questions, and major of Königsberg asked for his help. Euler was not pleased to receive this problem at first as he said “… I don’t believe this question has anything to do with mathematics.” Although after some time he believed that the problem was interesting enough to deal with.

Euler’s solution to Seven Bridges of Königsberg problem culminated a couple brand new mathematics branches to emerge: Topology and graph theory. What fascinates me about this story is that these branches came out almost 150 years after Euler’s solution.

Seven Bridges of Königsberg

Take any book on graph theory. Open first chapter’s first title. You’ll see Euler’s solution for this simple problem:

The old town of Königsberg has seven bridges as shown on the map. Is it possible to take a walk through the town, visiting each part of the town and crossing each bridge once and only once?

I encountered with this problem for the first time in the junior year of my math major. I first thought that the question was dull and I could solve it with ease. After spending a night with full of disappointments and a lot of cursing, I gave up. I knew that it was impossible to take that walk and I knew that it was possible when I removed one bridge. But I didn’t know why.

Soon enough, I realized that I needed fundamentals of the graph theory to understand the answer. Euler’s solution to this simple puzzle is the basis of a whole new mathematics branch: Graph theory.

What Euler Did

First thing Euler did was to imagine that bridges and lands as two different groups. According to him, solution of Königsberg problem didn’t depend on the physical appearances of them. Euler realized that it didn’t matter how long or curvy a bridge is, or if a land is an island or not.

He just thought that bridges could be represented as edges numbered from 1 to 7, and lands could be represented as vertices named A,B,C and D. When all Königsberg problem map is put into a piece of paper with using on edges and vertices, we would end up with a graph.

Degree of a vertex is the number of edges it has.

According to Euler, in order to find a solution for Königsberg problem either of these should be found in the graph:

1. Exactly two vertices should have odd number of degrees. That would mean one of the vertices is the start of the tour as the other is the finish.
2. All of the vertices should have even number of degrees. That would mean one could start the tour from any vertex.

Why?

If a vertex is neither start nor finish of the tour, then in case one edge comes to that vertex another one should leave so that the vertex would be neither the start nor the finish. (Because if an edge comes to a vertex and another doesn’t leave, it means that vertex is the finish of the tour. And if an edge leaves from a vertex but another doesn’t come back to it, then it means that vertex is the start of the tour.) This would mean that the vertex should have even number of degree. Although, if vertex has odd number of edges, in order to keep odd degree we shouldn’t let another edge to leave in case there is an edge coming to that vertex. This would mean that the so called vertex is either start or finish of the tour.

Also, if there are no vertices in the graph with an odd degree, then a chosen vertex could well be both start and finish of the tour.

Eulerian Path: If a path/tour in a graph contains each edge of the graph once and only once, then that path is called Eulerian Path.

I am aware that the paragraph above is a bit confusing. Let me give you a few examples to clear your mind about degree of a vertex.

Examples:

1. One Bridge: Two lands and one bridge graph.
   
2. Two Bridges: Two lands and two bridges graph.
   
3. Three Lands: Add land C into the previous graph.
   

In this case there are two vertices with odd number of degrees. Hence, there is a solution for this graph but vertices B and C should be the start and finish of the path.

Solution to Königsberg

Euler saw the Seven Bridges of Königsberg as this graph.

Numbers of degrees for the vertices are 5, 3, 3 and 3 in order. This means that the Königsberg graph has more than two odd numbers of vertices which is a violation for the rule. This is concludes that there is no such tour/path for the seven bridges of Königsberg.

Topology is the mathematical study of the properties that are preserved through twisting and stretching of objects. (Tearing, however, is not included.) Topologically a circle and an ellipse are equivalent to each other.

Check it out

Which objects are topologically equivalent to each other ?