Strong connectivity and directed triangles in oriented graphs. Partial results on a particular case of the Caccetta-Häggkvist conjecture

A particular case of Caccetta-Häggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order $n$ with minimum out-degree at least $0.3465n$ contains a directed cycle of length at most 3 (which currently is the best result). A weaker particular case says that a digraph of order $n$ with minimum semi-degree at least $1/3n$ contains a directed triangle. In a recent paper, by using the result of Kral et al, the author proved that for $β\geq 0.343545$, any digraph $D$ of order $n$ with minimum semi-degree at least $βn$ contains a directed cycle of length at most 3 (which currently is the best result). This means that for a given integer $d\geq 1$, every digraph with minimum semi-degree $d$ and of order $md$ with $m\leq 2.91082$, contains a directed cycle of length at most 3. In particular, every oriented graph with minimum semi-degree $d$ and of order $md$ with $m\leq 2.91082$, contains a directed triangle. In this paper, by using the result of Kral et al, we prove that every oriented graph with minimum semi-degree $d$, of order $md$ with $2.91082< m\leq 3$ and of strong connectivity at most $0.679d$, contains a directed triangle. This will be implied by a more general and more precise result, valid not only for $2.91082< m\leq 3$ but also for larger values of $m$. As application, we improve two existing results. The first result (Authors Broersma and Li), concerns the number of the directed cycles of length 4 of a triangle free oriented graph of order $n$ and of minimum semi-degree at least $\frac{n}{3}$. The second result (Authors Kelly, Kühn and Osthus), concerns the diameter of a triangle free oriented graph of order $n$ and of minimum semi-degree at least $\frac{n}{5}$.

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