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≥0.3465α≥0.3465, any digraph DD of order nn with minimum out-degree at least αnαn contains a cycle of length at most 3. Hamburger et al. (2007) (see [7]) proved that for β≥0.34564β≥0.34564, any digraph DD of order nn with both minimum out-degree and minimum in-degree at least βnβn contains a cycle of length at most 33. In this paper, by using the first result, we slightly improve the second bound. Namely, we prove that for β≥0.343545β≥0.343545, any digraph DD of order nn with both minimum out-degree and minimum in-degree at least βnβn contains a cycle of length at most 3. This result will be in fact a consequence of a quite general result.