Bipartite Graphs and Digraphs with Maximum Connectivity

Recently, some sufficient conditions for a digraph to have maximum connectivity or high superconnectivity have been given in terms of a new parameter which can be thought of as a generalization of the girth of a graph. In this paper similar results are derived for bipartite digraphs and graphs showing that, in this case, all the known conditions can be improved. As a corollary, it is shown that any bipartite graph of girth g and diameter D < g 2 (respectively D < g 1) has maximum vertex-connectivity (respectively maximum edge-connectivity). This implies a result of Plesnik and Z&n stating that any bipartite graph with diameter three is maximally edge-connected. 1. Notation and basic results In this paper G = (V,E) stands for a simple digraph, i.e. without loops or multiple edges, with set of vertices V = V(G) and set of (directed) edges E = E(G). If G is bipartite we will write V = VoU VI, where VO and VI denote the partite sets of vertices. If x E V, let T-(X) and r+(x) denote, respectively, the sets of vertices adjacent to and from x. Their cardinalities are the in-degree of x, 6-(x) = (r-(x)1, and the OUTdegree of x, 6+(x) = IT+(x)\. Th e minimum degree of G, 6 = 6(G), is the minimum over all the in-degrees and out-degrees of the vertices of G. For any pair of vertices x, y E V, a path xx1 x2 . . .x,-l y from x to y, with not necessarily different vertices, is called an x + y path. The distance from x to y is denoted by dG(x, y) = d(x, y), and D = D(G) = maxX,Y,V {d(x, y)} stands for the diameter of G. The distance from x to F c V, denoted by d&F), is the minimum over all the distances d(x,f), f E F. The distance from F to x, d(F,x), is defined analogously. We say that an x + y path avoids F if it contains no vertex of F. A digraph G = (V, E) is said to be (strongly) connected when for any pair of vertices x,y E V there always exists an x + y path. The connectivity (or vertex-connectivity) of G, JC = K(G), is the smallest number of vertices whose deletion results in a digraph that is either non-connected or trivial. The edge-connectivity of G, I = I(G), is defined analogously. Throughout the paper G * Corresponding author. 0166-218X/96/$15.00