Bipartite Connected Graph, Eulerian Circuit

Question

Give an example of a bipartite connected graph which has an even number of vertices and an Eulerian circuit, but does not have a perfect matching.

Answer 1

Bipartite ... Only Red and Blue vertices are joined.

An even number of vertices ... $3+5=8$.

Eulerian ... each vertex has even valency.

But ... there is clearly no matching.

Answer 2

Hint: In a bipartite graph, any edge in a matching must go from one half to the other. Using only this fact, can you think of a very simple criterion on a bipartite graph that will ensure that there is no perfect matching? Finding one such graph which also fulfills the other two requirements should be rather easy.