Abstract
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.
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