REMARKS ON HAMILTONIAN PROPERTIES OF POWERS OF DIGRAPHS

In the last thirty years many authors were engaged in considering the Hamiltonian behaviour of powers of undirected graphs. Started by Sekanina (1960), the investigations in this field have resulted in a lot of interesting and partly very profound propositions; for instance, the famous theorem of Fleischner (1974) on squares of blocks. The surprising thing is that in the case of directed graphs, however, no result of this kind has been published. In this note it is attempted to illustrate the reasons responsible for that different situation and to make a small first step in considering Hamiltonian properties of powers of digraphs.