Proving that vertex labeling is NP-complete
Given a graph $G$, consider its complement $G'$. The graph $G'$ is a yes instance of your problem if its vertices can be ordered so that adjacent vertices are not connected by an edge in $G'$. In other words, $G'$ is a yes instance of your problem if the vertices of $G$ can be ordered so that adjacent vertices are connected. So $G'$ is a yes instance of your problem if and only if $G$ has a Hamiltonian path.