Euler's Solution: The Degree of a Vertex

A Math Forum Project

Table of Contents:

Famous Problems Home

The Bridges of Konigsberg
· Euler's Solution
· Solution, problem 1
· Solution, problem 2
· Solution, problem 4
· Solution, problem 5

The Value of Pi
· A Chronological Table of Values
· Squaring the Circle

Prime Numbers
· Finding Prime Numbers

Famous Paradoxes
· Zeno's Paradox
· Cantor's Infinities
· Cantor's Infinities, Page 2

The Problem of Points
· Pascal's Generalization
· Summary and Problems
· Solution, Problem 1
· Solution, Problem 2

Proof of the Pythagorean Theorem

Proof that e is Irrational

Book Reviews

References

Links

Euler approached this problem by collapsing areas of land separated by the river into points, which he labelled with capital letters. Modern graph theorists call these vertices, and have gone on to represent them and bridges graphically.

For Konigsberg, let us represent land with red dots and bridges with black curves, or arcs:

Thus, in its stripped down version, the seven bridges problem looks like this:

   

The problem now becomes one of drawing this picture without retracing any line and without picking your pencil up off the paper. Consider this: all four of the vertices in the above picture have an odd number of arcs connected to them. Take one of these vertices, say one of the ones with three arcs connected to it. Say you're going along, trying to trace the above figure out without picking up your pencil. The first time you get to this vertex, you can leave by another arc. But the next time you arrive, you can't. So you'd better be through drawing the figure when you get there! Alternatively, you could start at that vertex, and then arrive and leave later. But then you can't come back. Thus every vertex with an odd number of arcs attached to it has to be either the beginning or the end of your pencil-path. So you can only have up to two 'odd' vertices! Thus it is impossible to draw the above picture in one pencil stroke without retracing.

The Generalization to Graph Theory

Euler went on to generalize this mode of thinking, laying a foundation for graph theory. Using modern vocabulary, we make the following definitions and prove a theorem:

Definition: A network is a figure made up of points (vertices) connected by non-intersecting curves (arcs).

Definition: A vertex is called odd if it has an odd number of arcs leading to it, other wise it is called even.

Definition: An Euler path is a continuous path that passes through every arc once and only once.

Theorem: If a network has more than two odd vertices, it does not have an Euler path.

Euler also proved this:

Theorem: If a network has two or zero odd vertices, it has at least one Euler path. In particular, if a network has exactly two odd vertices, then its Euler paths can only start on one of the odd vertices, and end on the other.

Problems

Problem 4:
For each of the networks below, determine whether it has an Euler path. If it does, find one.

figure 1 figure 2
figure 3 figure 4
figure 5 figure 6
Solution

Problem 5:
Without finding (or trying to find) an Euler path, determine whether the network below has any Euler paths.
figure 7
Solution
to the Bridges of Konigsberg
to the Value of Pi

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August, 1998