Let $\mathscr{ H}$ be a family of digraphs. A digraph $D$ is \emph{ $\mathscr{ H}$-free} if it contains no isomorphic copy of any member of $\mathscr{ H}$. For $k\geq2$, we set $C_{ \leq k}=\{ C_{ 2}, C_{ 3},\ldots,C_{ k}\}$, where $C_{ \ell}$ is a directed cycle of length $\ell\in\{ 2,3,\ldots,k\}$. Let $D_{ n}^{ k}(\xi,\zeta)$ denote the family of \emph{ ${ C}_{ \le k}$-free} strong digraphs on $n$ vertices with every vertex having out-degree at least $\xi$ and in-degree at least $\zeta$, where both $\xi$ and $\zeta$ are positive integers. Let $\varphi_{ n}^{ k}(\xi,\zeta)=\max\{ |A(D)|:\;D\in D_{ n}^{ k}(\xi,\zeta)\}$ and $\Phi_{ n}^{ k}(\xi,\zeta)=\{ D\in D_{ n}^{ k}(\xi,\zeta): |A(D)|=\varphi_{ n}^{ k}(\xi,\zeta)\}$. Bermond et al.\;(1980) verified that $\varphi_{ n}^{ k}(1,1)=\binom{ n-k+2}{ 2}+k-2$. Chen and Chang\;(2021) showed that $\binom{ n-1}{ 2}-2\leq\varphi_{ n}^{ 3}(2,1)\leq\binom{ n-1}{ 2}$. This upper bound was further improved to $\binom{ n-1}{ 2}-1$ by Chen and Chang\;(DAM, 2022), furthermore, they also gave the exact values of $\varphi_{ n}^{ 3}(2,1)$ for $n\in \{ 7,8,9\}$. In this paper, we continue to determine the exact values of $\varphi_{ n}^{ 3}(2,1)$ for $n\ge 10$, i.e., $\varphi_{ n}^{ 3}(2,1)=\binom{ n-1}{ 2}-2$ for $n\geq10$.
Categories: math.CO