# A condition that might force biregularity of a bipartite graph

Let $\Gamma$ be a finite connected bipartite graph with colour classes $U$ and $V$ such that:

• every vertex of $U$ has degree $n$, and $n\ge 3$;
• every vertex of $V$ has degree at least $4$;
• $\Gamma$ has diameter $4$;
• for every vertex $u$ of $U$, there are at least $(n+1)/2$ vertices in the neighbourhood $\Gamma(u)$ of $u$ that have degree the minimum degree of $\Gamma(u)$.

Is $\Gamma$ biregular (i.e., the vertices of $V$ have constant degree)? If not, then under what natural and weak conditions is biregularity forced to hold?