Superconnectivity of bipartite digraphs and graphs

A maximally connected digraph G is said to be super-κ if all its minimum disconnecting sets are trivial. Analogously, G is called super-λ if it is maximally arcconnected and all its minimum arc-disconnecting sets are trivial. It is first proved that any bipartite digraph G with diameter D is super-κ if D ≤ 2` − 1, and it is super-λ if D ≤ 2`, where ` denotes a parameter related to the number of short paths. These results allow us to show that if the order of a bipartite digraph G is big enough then superconnectivity is attained. For instance, if G is d-regular and has diameter D = 3 and ` ≥ 1, then G is super-λ if n > 4d; and if D = 4 and ` ≥ 2, then G is super-κ if n > 4d. In these cases the results are proved to be best possible. Similar results are given for bipartite (undirected) graphs. (For a graph it turns out that ` = (g − 2)/2, where g stands for the girth.)