A new bound for a particular case of the Caccetta–Häggkvist conjecture

In a recent paper, Hladký et al. (2009) (see [8]) proved that for α ≥ 0.3465 , any digraph D of order n with minimum out-degree at least α n contains a cycle of length at most 3. Hamburger et al. (2007) (see [7]) proved that for β ≥ 0.34564 , any digraph D of order n with both minimum out-degree and minimum in-degree at least β n contains a cycle of length at most 3 . In this paper, by using the first result, we slightly improve the second bound. Namely, we prove that for β ≥ 0.343545 , any digraph D of order n with both minimum out-degree and minimum in-degree at least β n contains a cycle of length at most 3. This result will be in fact a consequence of a quite general result.