I have a relatively simple question. I was given this exercise
A graph $G$ is called $2$–connected if for every pair of vertices $x$ and $y$ there are at least $3$ internally disjoint $xy$–paths in G. Show that every $3$–connected graph has an even cycle. (Hint: Use Menger’s theorem)
But if a graph is $3$-connected there is at least 3 internally disjoint paths between any $x$ and $y$. So $2$ will have to same parity. Take the union of these and the cycle is even.
Is this correct? I have not used the hint and it seems to be way too simple.