Mathematical Models of Euler's Circuits & Euler's Paths

Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

The mathematical models of Euler circuits and Euler paths can be used to solve real-world problems. Learn about Euler paths and Euler circuits, then practice using them to solve three real-world practical problems. Updated: 10/29/2021

Euler Paths and Circuits

In this video lesson, we are going to see how Euler paths and circuits can be used to solve real-world problems. You will see how the mailman and the salesman make use of these paths and circuits. You will also be presented with a seemingly complex problem to see if such a feat is possible. Keep watching!

First, let's quickly review what we know of Euler paths and circuits. An Euler path is a path in a graph where each edge is crossed exactly once. An Euler path can have any starting point with a different end point. A graph with an Euler path can have either zero or two vertices that are odd. The rest must be even.

An Euler circuit is a circuit in a graph where each edge is crossed exactly once. The start and end points are the same. All vertices must be even for the graph to have an Euler circuit. Let's get on with the problems now.

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  • 0:02 Euler Paths & Circuits
  • 1:12 Delivering the Mail
  • 2:48 The Traveling Salesman
  • 3:50 The 7 Bridges of Konigsberg
  • 5:56 Lesson Summary
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Delivering the Mail

Our first problem deals with delivering the mail. Imagine our postman with a stack of mail in his hand. He needs to deliver this mail to addresses on five different streets. He looks on the map, and he sees how the roads connect with each other. He draws a simplified version of the map. He marks each intersection with a number. Since he has a car, he would like to end at the same point where he began.

Euler model

To make the best use of his time, he needs to walk each road just once. Can he do so and if so, how?

Looking at his graph, we see that yes, it is possible to walk each road just once. He can park his car at intersection one, walk to intersection two, then three, then four, then five, and then back to one. Doing this, he will have walked each road only once. Since he ended up where he began, he actually walked an Euler circuit.

Are there other ways he could have done an Euler circuit? Yes. There are actually ten different Euler circuits he could have taken. He could have started at point one, gone to point five, then four, three, two, and then back to one again. He can actually begin at any one of the points and go either way. We have five points and two ways to go from each point. This makes for a total of ten possible routes he could have taken; ten possible Euler circuits.

The Traveling Salesman

Now, let's look at the traveling salesman problem. His is similar to the postman's. Since our salesman travels, he doesn't mind ending up at a different point from where he began. But just like the postman, he wants to make best use of his time and travel each road just once. This salesman did what the postman did and drew a simplified version of the roads he wants to travel.

Euler model

He also has five roads to reach. Can he go through all five roads just once? Yes, he can. If he begins at point three, he can then go to point two, one, five, four, and then one again. He will have visited all five roads just once. Since he ended up at a different spot from where he began, he has traveled an Euler path. Are there more Euler paths that our salesman could have taken? Yes, there are. Can you spot them?

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Practice:
Mathematical Models of Euler's Circuits & Euler's Paths Quiz

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