All 2-regular graphs with the same number of vertices are isomorphic to each other.

Question

I need to prove or disprove that all 2-regular graphs with the same number of vertices are isomorphic to each other. I've tried to come up with a counter example but it seems wrong and forced to me. Below you can see two edgelists for two graphs who are both 2-regular but not isomorphic to each other. $ E_{1} := \{(a,b),(b,c),(c,d),(d,e),(e,f),(f,g),(g,a) \}$ $ E_{2} := \{(a,b),(b,c),(c,a),(d,e),(e,f),(f,g),(g,d) \}$

is the second edgelist still a graph or two distinct graphs?

In general, how do I proceed on proofs like this? always try to find a counter example or how?

Thank you.

Answer 1

That's right, the first is a connected graph, the second is a graph that isn't connected.

What is true is that all connected $2$-regular graphs with $n$ vertices are isomorphic, being $n$-cycle graphs.

Answer 2

Your counterexample is correct, because there was no condition about connectivity.