SOME REMARKS ON ADAM CONJECTURE FOR SIMPLE DIRECTED-GRAPHS

The classes of multidigraphs for which Adam's conjecture (that any digraph containing a directed cycle has an arc whose reversal decreases the total number of directed cycles) does not hold, were described in Grinberg (1988), Jirasek (1987) and Thomassen (1987). The question remains, however, open for simple directed graphs. In the paper we show that the conjecture holds for all simple digraphs containing a nontrivial strongly connected component which is not strongly 2-connected and for simple digraphs that become acyclic after reversal of at most three of their arcs.