At most how many edges can a connected bipartite graph with $n$ vertices in each class can have so that there is no perfect matching?
If we omit the connectedness condition, then the maximum is $n(n-1)$ ($K_{n,n-1}$ with an isolated vertex is an example; the upper bound is proven by induction - if we assume there are at least $n^2 - n + 1$ edges, then there is a vertex of degree $n$; removing it and a neighbour of degree at most $n-1$ does the job).
However, in the connected case I have no idea even for an answer. Any help appreciated!