Eulerian Circuits and Eulerian Graphs | Graph Theory, Euler Graphs and Euler Circuits

What are Eulerian graphs and Eulerian circuits? Euler graphs and Euler circuits go hand in hand, and are very interesting. We’ll be defining Euler circuits first in today’s lesson, as well as showing an example of why these circuits might be interesting to begin with, then we go into Euler graphs, and discuss how to describe an Euler circuit and how to know if a graph is an Eulerian graph!

An Eulerian circuit in a graph G is a circuit that visits every edge of G. Remember that a circuit is a closed walk that repeats no edges. A graph doesn’t have to be connected to have an Eulerian circuit, since it could have a disconnected but isolated vertex, which has no edges incident to it that need to be traversed by an Eulerian circuit. However, for a graph to have an Eulerian circuit, one necessary condition is that it has exactly one non-trivial component. So, any disconnected graph with an Euler circuit is only disconnected because it has some trivial components. Since that isn’t very interesting we usually just discuss connected graphs when talking about Eulerian circuits.

The good thing is that if you understand what an Eulerian circuit is, you understand Eulerian graphs! A connected graph is Eulerian if it contains an Eulerian circuit. What we find in the video lesson is that our Eulerian graph has vertices all of even degree. Interestingly, a connected graph is Eulerian if and only if all of its vertices have even degrees. So if all of a connected graph’s vertices have even degrees, then it has an Eulerian circuit, and if a connected graph has an Eulerian circuit, then all of its vertices have even degrees! This is very fascinating, and I encourage you to try to prove it on your own! Let me know if you’d like to see me go over a proof in another lesson!

The graph I use in this lesson is straight out of the textbook that introduced me to Graph Theory, “A First Course in Graph Theory“ by Gary Chartrand and Ping Zhang. It’s a wonderful text! You can purchase this book through my Amazon affiliate link below! Using the affiliate link costs you nothing, and helps me continue to work on Wrath of Math!

PURCHASE "A First Course in Graph Theory": https://amzn.to/31hgvvJ

Eulerian circuits are also sometimes called Euler tours and Euler cycles.

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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