A new bound for a particular case of the Caccetta-Haggkvist conjecture

In a recent paper, Hladky et al. (2009) (see [8]) proved that for alpha >= 0.3465, any digraph D of order n with minimum out-degree at least an contains a cycle of length at most 3. Hamburger et al. (2007) (see [7]) proved that for beta >= 0.34564, any digraph D of order n with both minimum out-degree and minimum in-degree at least beta 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 beta >= 0.343545, any digraph D of order n with both minimum out-degree and minimum in-degree at least beta n contains a cycle of length at most 3. This result will be in fact a consequence of a quite general result. (C) 2010 Elsevier B.V. All rights reserved.