Fleury's Algorithm for Finding an Euler Circuit

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  • 0:02 A Graph
  • 0:49 Fleury's Algorithm
  • 2:29 Start Point
  • 2:59 Choosing Edges
  • 4:21 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.

In this video lesson, you will learn a method for finding an Euler circuit. Learn the one criterion that is the basis for all your decisions when choosing a route.

A Graph

Your friend is taking a math class that talks about graph theory. He has a problem, though, and because he knows that you have been reading up on graph theory, he decides to ask you about his problem.


He needs to know how he can find an Euler circuit given this graph. He already knows that one exists, that there is a path where all the edges are passed exactly once and you end up where you began. The problem is he doesn't know how to go about finding such a path. He looks at all the vertices and edges and just gets confused.

So, can you help? Yes, you certainly can. It's a good thing that you are watching this video lesson because it is in this video lesson that you will learn a method for finding an Euler circuit given a graph.

Fleury's Algorithm

You start explaining to your friend. The method that we will be using is called Fleury's algorithm. It is a very simple algorithm when you look at it. But, it works for any graph with an Euler circuit or path in it. Remember that an Euler path is when you cross all edges just once but you end up in a different spot. An Euler circuit is the same as an Euler path except you end up where you began.

Fleury's algorithm shows you how to find an Euler path or circuit. It begins with giving the requirement for the graph. The graph must have either 0 or 2 odd vertices. An odd vertex is one where the number of edges connecting the vertex to other vertices is odd. If we have 2 odd vertices, then we will have an Euler path. If we have 0 odd vertices, then we will have an Euler circuit. If we have 2 odd vertices, then we start at one of those two vertices. If we have 0 odd vertices, then we can start anywhere.

To find our way, we choose the edge that is not a bridge if we have a choice. If we don't, then we cross the bridge. A bridge can be likened to a real-world bridge that connects an island to the mainland. So a bridge in a graph is an edge that connects one part of the graph to the rest. So, if you are at a vertex where one edge is a bridge and the other is not, choose the one that is not a bridge. Remember the phrase, 'Don't burn your bridges.' Keep going until you have crossed each edge. Once we have crossed each edge, we can pretend that the edge is no longer there.

Start Point

Let's see how Fleury's algorithm plays out with the graph that your friend has presented you. This graph has 0 odd vertices. According to Fleury's algorithm, then, we can start at any point that we want. Let's start at the vertex that is at the way bottom. Because we have 0 odd vertices, we also know that what we will get is an Euler circuit. If we had 2 odd vertices in our graph, we would be finding an Euler path instead.

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