Pham 1

Clein Pham

Dr. Shanyu Ji

Math 4388

The Seven Bridges of Königsberg

There are many who say the phrase, “cross the when you get to it”. Meaning, you should address an issue when it actually occurs. But what if crossing the bridge itself was the issue? A famous problem in mathematics is the Seven Bridges of Königsberg. Königsberg was a city in Prussia, currently known as Kaliningrad in Russia. The Pregal River split the city in two which included two large islands, Kneiphof and Lomse, all of which were connected via seven bridges. The problem in question was whether or not it was possible to travel throughout the city that would entail the crossing of each bridge only once. The problem was resolved by mathematician in 1736, and his resolution would lay the foundation for and became the basis for the idea of .1

1 Shields, Rob. Cultural Topology: The Seven Bridges of Königsburg. Theory, Culture &

Society. 2012. Pham 2

Euler was born in the 1700s and made various significant and influential contributions to mathematics, specifically calculus, graph theory, and topology.2 Euler is the only mathematician to have not one, but two numbers named after him: Euler’s number e and Euler’s constant . His solution to the problem of the Seven Bridges of Königsberg led to the development of graph theory and also the following equation:

V − E + F = 2 which is related to topology. Let’s take a look at his solution to the problem in more detail.

Euler first addressed the problem by stating that the choice of route inside each land mass is irrelevant. The only thing that mattered when it came to routes was the order of the bridges crossed. This allowed Euler to think about things in abstract terms, calling each land mass a

“vertex” or node and each bridge a “edge”. Only the connection between the masses is relevant, so whether or not an edge is present between nodes is important.

Euler then observed that excluding the endpoints, when one enters a node via a bridge, they must exit that same node via another bridge. Meaning throughout the walk, the number of times one enters a non-terminal vertex must equal the number of times one leaves it. If every bridge is traveled across exactly once, then there must be an even number of bridges for every land mass (except for the ones chosen for start and finish). Here we enter a roadblock: every land mass in Königsberg had an odd number of bridges touching it.

2 Dunham, William. Euler: The Master of Us All. Mathematical Association of America.

1999. Pham 3

Euler defined a of a node as the number of edges touching it. Euler showed that it was necessary to have exactly zero or two nodes of odd degree for this specific walk. This would be eventually called a Eulerian , in which a trail must visit every edge exactly once.3 If nodes with an odd degree are present, then then any Eulerian path will begin at one and end at another.

A graph of Königsberg’s seven bridges has four nodes all of odd degree (one node with degree 5 and three nodes with degree 3), therefore there is no Eulerian path and thus the problem of the

Seven Bridges of Königsberg has no solution.

The significance of this “solution”, or lack of thereof, has led to the development of graph theory and is thought to be the first true proof in the theory of networks, known now as combinatorics.4 In modern terms, graph theory relates paths and edges and states that if there is a path of a that crosses an edge only one time, then there exists at most two vertices of odd degree. Meaning, a multigraph occurs when two vertices are connected by more than one edge, a la the Bridges of Königsberg. Furthermore, if the path is a loop (the path begins and ends at the same vertex), then there will be no vertices with an odd degree. This is the basis for

3 Biggs, N. L., Lloyd, E. K., Wilson, R. J. Graph Theory 1736-1936. Clarendon Press,

Oxford. 1976.

4 Newman, M. E. J. The structure and function of complex networks. Department of

Physics, University of Michigan. 2003. Pham 4 modern graph theory, and the influence of Euler’s solution led to the development of a branch of mathematics known as topology.

Topology deals with geometric properties of objects that have undergoes physical irregularities such as stretching, twisting, or crumbling. Euler’s polyhedron formula (stated above) relates the number of vertices, edges, and faces of a polyhedron, with many regarding this analysis as the birth of topology.5 A key idea from this is that topology is not necessarily concerned with the shape of an object, but rather features of the object.

Euler’s work was presented and published as Solutio problematis ad geometriam situs pertinentis (The solution of a problem relating to the geometry of position) in 1735. Currently, only five of the original seven bridges remain present, as two were destroyed in the Second

World War.6 It is interesting to note that as five of the bridges still remain, in terms of graph theory this now leaves two nodes with degree 2 and the other two nodes with degree 3, making a

Eulerian path possible.7 Euler’s solution has allowed us to find an answer to the problem (by removing two bridges) and contributed greatly to other sectors of mathematics.

5 Richeson, D. Euler’s Gem: The Polyhedron Formula and the Birth of Topology.

Princeton University Press. 2008.

6 Taylor, Peter. What Ever Happened to Those Bridges. Australian Mathematics Trust.

2000.

7 https://people.engr.ncsu.edu/mfms/SevenBridges/ Pham 5

Works Cited

Shields, Rob. Cultural Topology: The Seven Bridges of Königsburg. Theory, Culture & Society.

2012.

Dunham, William. Euler: The Master of Us All. Mathematical Association of America. 1999.

Cromwell, Peter R. Polyhedra. Cambridge University Press. 1999.

Biggs, N. L., Lloyd, E. K., Wilson, R. J. Graph Theory 1736-1936. Clarendon Press, Oxford.

1976.

Newman, M. E. J. The structure and function of complex networks. Department of Physics,

University of Michigan. 2003.

Richeson, D. Euler’s Gem: The Polyhedron Formula and the Birth of Topology. Princeton

University Press. 2008.

Taylor, Peter. What Ever Happened to Those Bridges. Australian Mathematics Trust. 2000. https://people.engr.ncsu.edu/mfms/SevenBridges/