Connectivity of large bipartite digraphs and graphs

This paper studies the relation between the connectivity and other parameters of a bipartite (di)graph G. Namely, its order n, minimum degree delta, maximum degree Delta, diameter D, and a new parameter l related to the number of short paths in G. (When G is a bipartite undirected - graph this parameter turns out to be l=(g-2)/2, where g stands for its girth.) Let n(Delta, l)=1 + Delta + Delta(2) + ... + Delta(l). As a main result, it is shown that if n > (delta - 1){n(Delta, l) + n(Delta, D-l-1)-2} + 2, then the connectivity of the bipartite digraph G is maximum. Similarly, if n > (delta-1){n(Delta, l) + n(Delta, D-l-2)}, then the arc-connectivity of G is also maximum. Some examples show that these results are best possible. Furthermore, we show that analogous results, formulated in terms of the girth, can be given for the undirected case.