Longest-path rotation

If an endpoint of a longest path is adjacent to an internal vertex, replacing the incident path edge by that chord produces another longest path with the same vertex set and a new endpoint. Every neighbour of every endpoint obtained this way must remain on the original path.

This rotation technique is used in arguments surrounding Pósa's theorem and other Hamiltonian cycle criteria.

Let $A$ be $k$ vertices of a bipartite graph, each of degree at least $k$. If the graph has no $4$-cycle, then $|N(A)|\ge 2k-1$. Indeed, any pair in $A$ has at most one common neighbour, soIf $|N(A)|\le 2k-2$, Cauchy--Schwarz and ${\sum }_y{d}_A(y)\ge k^2$ make the left side strictly larger than $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$, a contradiction.