论文标题
$ c _ {\ leq k} $的最大尺寸至少两个，至少两个
Maximum size of $C_{\leq k}$-free strong digraphs with out-degree at least two
论文作者
论文摘要
令$ \ mathscr {h} $为Digraphs家族。如果它包含$ \ Mathscr {h} $的任何成员的同构副本，则DIGRAPH $ D $ IS \ EMPH {$ \ MATHSCR {H} $ - free}。对于$ k \ geq2 $，我们设置$ c _ {\ leq k} = \ {c_ {2}，c_ {3}，\ ldots，c_ {k} \} $，其中$ c _ {\ ell} $是$ \ ell \ ell \ ell \ ell \ in \ in $ in \ in $ in \ ld} $。令$ d_ {n}^{k}（ξ，ζ）$表示\ emph {$ {c} _ {\ le k} $ - free} $ n $ vertices上的强度}的家族，每个顶点至少都有$ n $ deg的$ n $ vertices，至少在$ $ $ $ $ $上，$ n $ n $ n of degebree，至少均为$ c。令$ $φ_{n}^{k}（ξ，ζ）= \ max \ {| a（d）|：\; d \ in d_ {n}^{n}^{k}（酮，ζ）\} $和$ $ $ $ $ $ d_ {n}^{k}（ξ，ζ）：| a（d）| =φ_{n}^{k}（ξ，ζ）\} $。 Bermond等人\;（1980）证实了$φ_{n}^{k}（1,1）= \ binom {n-k+2} {2} {2}+k-2 $。 Chen and Chang \;（2021）表明$ \ binom {n-1} {2} -2 \ leq或leq或leq或此外，这种上限进一步改善到$ \ binom {n-1} {2} -1 $，由Chen和Chang \;（Dam，2022），此外，他们还提供了$ n \ in \ in \ in \ in \ {7,8,8,8,9 \ \ n \ in \ in \ in \ in \ in \ in \ in \ n \ in \ n \ in {7,8,8,9 \ \ 9 \ \ \ n \。在本文中，我们继续确定$φ_{n}^{3}（2,1）$的确切值，$ n \ ge 10 $，即$φ__{n}^{3}^{3}（2,1）= \ binom {n-binom {n-1} {2} {2} {2} {2} -2 $ n \ n \ geq10 $。
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 $D_{n}^{k}(ξ,ζ)$ denote the family of \emph{${C}_{\le k}$-free} strong digraphs on $n$ vertices with every vertex having out-degree at least $ξ$ and in-degree at least $ζ$, where both $ξ$ and $ζ$ are positive integers. Let $φ_{n}^{k}(ξ,ζ)=\max\{|A(D)|:\;D\in D_{n}^{k}(ξ,ζ)\}$ and $Φ_{n}^{k}(ξ,ζ)=\{D\in D_{n}^{k}(ξ,ζ): |A(D)|=φ_{n}^{k}(ξ,ζ)\}$. Bermond et al.\;(1980) verified that $φ_{n}^{k}(1,1)=\binom{n-k+2}{2}+k-2$. Chen and Chang\;(2021) showed that $\binom{n-1}{2}-2\leqφ_{n}^{3}(2,1)\leq\binom{n-1}{2}$. 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 $φ_{n}^{3}(2,1)$ for $n\in \{7,8,9\}$. In this paper, we continue to determine the exact values of $φ_{n}^{3}(2,1)$ for $n\ge 10$, i.e., $φ_{n}^{3}(2,1)=\binom{n-1}{2}-2$ for $n\geq10$.
