Let $G$ be a 3-connected graph. Prove that for every three vertices $a, b, c$ of $G$ there exists a cycle in $G$ that contains $a,b$ but not $c$.
Here is my work. Since $G$ is a 3-connected graph, the minimum edges need to be removed in order to make it become disconnected is $3$. The degree of very vertex must be at least $3$. If we remove edge $ac$ and $bc$ if they exist, it's still connected. Then, by removing the other edges that connected to $c$, the graph is still connect. Since the degree of vertices $a$ and $b$ are at least 2, there must be a cycle contains $a$ and $b$.
Is it correct?