Let $G$ be a bipartite graph with $n$ vertices. Prove that if every vertex has degree at least $\frac{n}{4} + 1$, then $G$ is connected.
I'm assuming that number of vertices in this bipartite graph must be divisible by 4, not sure where to go from there.
Let the two parts of the graph be $A$ and $B$ if it is not connected then it contains $k\geq2$ connected components. Each of these connected components contains at least $\frac{n}{4}+1$ vertices in $A$ and at least $\frac{n}{4}+1$ vertices in $B$. So each connected component has at least $\frac{n}{2}+2$ vertices. So if we have $k$ connected components we need $\frac{kn}{2}+2k$ vertices, even for $k=2$ we get $n+4$ vertices. So it is impossible for $G$ to be disconnected.