Euler's Theorems: Circuit, Path & Sum of Degrees

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  • 0:01 A Graph
  • 0:45 Euler's Circuit Theorem
  • 1:50 Euler's Path Theorem
  • 2:41 Euler's Sum of Degrees Theorem
  • 3:44 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson, and you will understand how Euler's circuit theorem, Euler's path theorem, and Euler's sum of degrees theorem will help you analyze graphs. Also, get some practice with the quiz.

A Graph

In this video lesson, we will go over three of Euler's theorems relating to graph theory. Why are these important? These are important because they help you to analyze graphs such as this one:

Euler theorem

Why are graphs such as these important? If our graph represents a neighborhood where the dots are intersections and the lines are the roads, then graph theory can help us find the best way to get around town. For example, graph theory can help the mailman deliver his mail so that he doesn't have to back-track or pass by the same road twice. Euler's theorems come in handy because they tell the mailman whether an efficient route is even possible just from looking at the graph. How does this work? Let's take a look at Euler's theorems and we'll see.

Euler's Circuit Theorem

The first theorem we will look at is called Euler's circuit theorem. This theorem states the following: 'If a graph's vertices all are even, then the graph has an Euler circuit. Otherwise, it does not have an Euler circuit.' What does this mean for our mailman?

Recall that an Euler circuit is a route where you can pass by each edge or line in the graph exactly once and end up where you began. This helps the mailman figure out whether there is a route that he can take where he ends up where he began and where he goes through each road just once. If such a route doesn't exist in the first place, then there is no point for him to even try to figure one out. Our vertices are of even degree if there is an even number of edges connecting it to other vertices.

If all vertices have an even degree, the graph has an Euler circuit
Graph of mail route

Looking at our graph, we see that all of our vertices are of an even degree. The bottom vertex has a degree of 2. All the others have a degree of 4. This means that the graph does have an Euler circuit. This tells the mailman that, yes, there does exist a route where he doesn't have to back-track. He can now go about figuring out such a route.

Euler's Path Theorem

This next theorem is very similar. Euler's path theorem states the following: 'If a graph has exactly two vertices of odd degree, then it has an Euler path that starts and ends on the odd-degree vertices. Otherwise, it does not have an Euler path.'

Recall that an Euler path is a path where you pass by each edge or line in the graph exactly once, and you end up in a different spot than where you began. A path is very similar to a circuit, with the only difference being that you end up somewhere else instead of where you began. An Euler path is good for a traveling salesman or someone else who doesn't need to end up where he began. Looking at our graph, we see that we don't have any vertices that are odd. This tells us that this graph does not have an Euler path in it.

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