Maximum size of $C_{\leq k}$-free strong digraphs with out-degree at least two

TL;DR

This paper determines the maximum number of edges in certain strong digraphs that avoid small directed cycles, with specific degree constraints, extending previous bounds to exact values for larger graphs.

Contribution

It establishes the exact maximum edge count for $C_{ ext{leq} 3}$-free strong digraphs with out-degree at least two for all $n \

Findings

Maximum edges for $n ext{ } extgreater= 10$ is $inom{n-1}{2}-2$.

Extended previous bounds to exact values for larger $n$.

Provides a complete characterization for the maximum size of these digraphs.

Abstract

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 $n\geq10$.