# "Give an example of a graph whose vertices are all of even degree, which does not contain a Eulerian Path"

I was given the above question in a test. I really struggle to see how it is possible that a Eulerian trail/path can fail to exist in a graph with all even degree vertices. When you consider that a Eulerian Cycle exists in any connected graph whose vertices are all of even degree, it makes it even more difficult to consider.

Is there an easy way to find such a graph or am I missing something? E.g. should I be considering, say, a graph consisting of disjoint sets of vertices, such as a graph of 8 vertices, with two sets of 4 vertices connected to make 2 disjoint squares (hence all vertices would have degree of 2)? Does this count? Any help appreciated!

• "Does this count?" Of course, such graph will be the answer to your question! Commented May 5, 2022 at 3:02

Take any $$\textbf{connected}$$ graph $$H$$ that have all degrees being even and let $$G$$ be $$H$$ union an isomorphic copy of it. $$G$$ is the graph you want.