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.

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.