Some remarks on Ádám's conjecture for simple directed graphs

Jirasek, J., Some remarks on Adam’s conjecture for simple directed graphs, Discrete Mathematics 108 (1992) 327-332. 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 of their arcs. For every digraph G = (V, A) containing a directed cycle there is an arc (x, y ) E A whose reversal decreases the total number of its cycles. The conjecture can also be formulated by means of the number of directed most three paths. Let (x, y) denote the number of the paths from the vertex x to the vertex y. Then the number of the cycles containing the arc (x, y ) is just (y, X) and after its reversal there will be (x, y) 1 cycles containing this reversed arc. Hence Adam’s conjecture can be formulated as follows: For every non-cycle-free digraph G there is an arc (x, y ) such that (x, y) 1~ (y, x). Now, it can be easily seen that the following propositions hold. 0012-365X/92/$05.00