Superconnectivity of bipartite digraphs and graphs Original Research Article

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 arc-connected 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 ⩽ 2ℓ − 1, and it is super-λ if D ⩽ 2ℓ, where ℓ 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 ℓ ⩾ 1, then G is super-λ if n > 4d; and if D = 4 and ℓ ⩾ 2, then G is super-κ if n > 4d2. 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 ℓ = (g − 2)/2, where g stands for the girth.)