Turán number of 3-free strong digraphs with out-degree restriction

Abstract

A digraph D is r-free if such D has no directed cycles of length at most r, where r is a positive integer. In 1980, Bermond et al. showed that if D is an r-free strong digraph of order n, then the size of D is at most n−r+22+r−2 and the upper bound is tight, for r≥2. Namely, they determined the Turán number of Cr-free strong digraphs of order n, where Cr={C2,C3,…,Cr} and Ci is a directed cycle of length i∈{2,3,…,r}. Specially, for r=3, the maximum size of 3-free strong digraphs of order n is n−12+1. Let Φn(ξ,γ) be a family of 3-free strong digraphs of order n in which the minimum out-degree is at least ξ and the minimum in-degree is at least γ, where both ξ and γ are positive integers. Let φn(ξ,γ) be the maximum size of digraphs of Φn(ξ,γ), i.e., Turán number of 3-free strong digraphs of order n with out and in-degree restrictions. We denote ϕn(ξ,γ)={D∈Φn(ξ,γ):thesizeofDisequaltoφn(ξ,γ)}. Recently, Chen et al. described ϕn(1,1), i.e., all 3-free strong digraphs of order n with size n−12+1. They also gave the bound of φn(2,1), that is, n−12−2≤φn(2,1)≤n−12. In this paper, we improve the upper bound of φn(2,1) to n−12−1 and we thus get n−12−2≤φn(2,1)≤n−12−1. In addition, we show that φn(2,1)=n−12−1 by constructing two 3-free strong digraphs with minimum out-degree two whose size reaches n−12−1 for n=7,8, and verify φn(2,1)=n−12−2 for n=9. As a consequence, Turán number of 3-free strong digraphs of order n with out-degree at least two, i.e., φn(2,1), is one of n−12−1 and n−12−2.