Eulerian Graphs

Eulerian graphs are mostly used to solve some practical problems. One major type of problems occur with Eulerian graphs, they are called Traveler's Problems.

An Explorer's problem asks questions like:

This type of problem focuses on finding a closed trail which includes every edge of the graph. This is actually the definition of a Eulerian graph.

Another way to think of a Eulerian graph is asking whether or not the graph can be drawn without retracing edges or lifting the pencil from the paper

(Scaff, 2017)

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Think back to the section about the beginning of graph theory, and Königsberg bridges problem. This problem asks to find a route crossing each bridge exactly once, this corresponds exactly to that of finding an Eulerian trail. Thus, the Königsberg bridges problem is an application of Eulerian trails. If you followed the links and tried to solve this puzzle for yourself, you know that there is no Eulerian trail that exists. A breakdown of Euler's proof of this is presented in

There are other types of problems that are also Eulerian graphs. These other types are

A Eulerian graph is created when the number of points is odd, because they can be drawn in one continuous stroke as shown below.

There are a few other types of problems, which are listed below.Research each of these on your own and try to decide why it is useful to use Eulerian graphs with them.