This paper studies the bipartite graphs G in which $\alpha _{\text{BIP}} ( G )$, the maximum order of an induced balanced bipartite subgraph without edges, is equal to 2. When its order is at least 10, it is shown that G contains a Hamiltonian path, provided that it is connected, and that, if its minimum degree is at least 2, then it is bipancyclic. Similar results concerning the bipartite digraphs D in which $\alpha _{\text{BIP}}^2 ( D )$ are given, and the maximum order of an induced balanced bipartite subdigraph without 2-cycles, is equal to 2.

Abstract The mean distance of a simple connected graph G of order n is defined by magnified image We answer a problem of Plesnik about the edge‐vulnerability of G related to μ, i.e., we find a bound for μ(G − e) − μ(G) and for magnified image where e is an edge of G such that G − e is still conencted. This question is of interest in interconnection La distance moyenne d'un graphe G, simple et connexe, d'ordre n, est définie par magnified image Nous répondons ici à un problème de Plesnik sur l'arěte‐vulnérabilité de G par rapport à μ, i.e. nous trouvons une borne pour μ(G −e) − μ(G) et pour magnified image oú e est une arěte de G telle que G − e soit encore connexe.