# Non isomorphic graphs with closed eulerian chains

I need to construct 2 graphs that are non isomorphic and have 3 of the following properties.

• Same number of vertices
• Same number of edges
• Both contain a closed eulerian path

I was thinking of the graph invariant to be "coherent components". With 1 graph containing 1 coherent component and the other one 2. But I couldn't get the number of edges the same.

Here's what I've tried:

2 Graphs

Does anyone know any examples of such graphs with these 3 properties, the invariant can also be something else.

Best regards

Each graph here has $6$ vertices, $8$ edges, and an Eulerian circuit. Each also has two vertices of degree $4$, but on the left, the two aren't adjacent, and on the right, they are, hence these graphs are nonisomorphic. I'm not sure what your observation about components was, but these two are connected, so it seems that that idea won't apply here. EDIT: Some more examples for graphs on $8$ vertices, $10$ edges. It's also worth noting that the last example has a different degree sequence from the others.  