3-Free Strong Digraphs with the Maximum Size

Abstract

Bermond et al in 1980 proved that if the size of a strong digraph D of order n is at least $\left( {\begin{array}{c}n-k+2\\ 2\end{array}}\right) +k-1$, $k\ge 2$, then the girth of D is no more than k. Consequently, when D is a 3-free strong digraph of order n without loops or parallel arcs, which means that every directed cycle in D has length at least 4, the maximum size of D is $\left( {\begin{array}{c}n-1\\ 2\end{array}}\right) +1$. In 2008, Seymour et al proved that if D is a 3-free digraph, then $\beta (D)\le \gamma (D)$, and they further conjectured that $\beta (D)\le \frac{1}{2}\gamma (D)$ for every 3-free digraph D, where $\beta (D)$ denotes the size of the smallest subset $X\subseteq A(D)$, such that $D\backslash X$ is acyclic, and $\gamma (D)$ is the number of unordered pairs $\{u,v\}$ of vertices such that u, v are nonadjacent in D. In this paper, we first describe all 3-free strong digraphs of order n with the maximum size $\left( {\begin{array}{c}n-1\\ 2\end{array}}\right) +1$. Then we prove that such 3-free strong digraphs satisfy the minimum out-degree of D equals 1 and $\beta (D)\le \frac{1}{2}\gamma (D)$. Moreover, we prove that if the minimum out-degree of a 3-free strong digraph D is at least 2, then the maximum size of D is between $\left( {\begin{array}{c}n-1\\ 2\end{array}}\right) -2$ and $\left( {\begin{array}{c}n-1\\ 2\end{array}}\right)$.