Can anyone give me an illustration of a graph that satisfies the following theorem concerning connected graphs?

Question

A connected graph has an Eulerian walk if and only if the number of odd vertices in it is either 0 or 2.

Note: Eulerian path and Eulerian walk is the same. My prof uses Eulerian walk that's why that is what I have written.

Thank you so much!

Answer 1

As pointed out in the comments, all connected graphs satisfy the theorem as they either: a) have 2 or 0 odd degree vertices and an Eulerian path, or b) have 1 or 3+ odd-degree vertices and don't have a Eulerian path.

The housegraph example given has exactly one vertex of odd degree so it doesn't have a Eulerian path.

Strangely, the definition given on Wolfram mathworld :

A connected graph has an Eulerian path iff it has at most two graph vertices of odd degree.

is wrong, as it would include graphs with a single vertex of odd degree.