Here is my proof:
Suppose $ G_1 = (V_1, E_1) $ and $G_2 = (V_2, E_2)$.
Then by the premise $$ \exists C_1 : i_1, i_2, \cdots, i_n=i_1, \forall i \in V_1$$ And $$ \exists C_2 : j_1, j_2, \cdots, j_n=j_1, \forall j \in V_2$$ where C1 and C2 are cycles.
Then there exists a bijection $ f: V_1 \rightarrow V_2, \hspace {3 pt} $ $f(i_n) = j_n$ and such that if $(i_n, i_{n+1}) \in E_1$ then $(f(i_n), f(i_{n+1})) \in E_2.$
So $G_1$ and $G_2$ are isomorphic $\Box$
It is my first time doing proofs on graphs and it feels a little weird and unsure.
Can you check for possible mistakes? Or any advice? Thanks!