Proof: Graph is Eulerian iff All Vertices have Even Degree | Euler Circuits, Graph Theory

A nontrivial connected graph is Eulerian if and only if every vertex of the graph has an even degree. We will be proving this classic graph theory result in today's lesson!

Remember that an Eulerian graph is a graph containing an Eulerian circuit, check out my lesson on them if you need a recap:    • Eulerian Circuits and Eulerian Graphs...  

An Eulerian circuit is a circuit containing every edge of a graph.

The first direction of the proof is pretty easy, we show that a Eulerian graph's vertices all have even degrees. The other direction, proving that if all vertices of a nontrivial connected graph have even degree then the graph contains an Eulerian circuit, takes a bit more work. We will use a bunch of proofs by contradiction and find that the theorem is fairly straightforward to prove, even if it does take a lot of work!



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

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