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- W4308613871 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 $kgeq2$, we set $C_{leq k}={C_{2}, C_{3},ldots,C_{k}}$, where $C_{ell}$ is a directed cycle of length $ellin{2,3,ldots,k}$. 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 $varphi_{n}^{k}(xi,zeta)=max{|A(D)|:;Din D_{n}^{k}(xi,zeta)}$ and $Phi_{n}^{k}(xi,zeta)={Din D_{n}^{k}(xi,zeta): |A(D)|=varphi_{n}^{k}(xi,zeta)}$. Bermond et al.;(1980) verified that $varphi_{n}^{k}(1,1)=binom{n-k+2}{2}+k-2$. Chen and Chang;(2021) showed that $binom{n-1}{2}-2leqvarphi_{n}^{3}(2,1)leqbinom{n-1}{2}$. 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 $nin {7,8,9}$. In this paper, we continue to determine the exact values of $varphi_{n}^{3}(2,1)$ for $nge 10$, i.e., $varphi_{n}^{3}(2,1)=binom{n-1}{2}-2$ for $ngeq10$." @default.
- W4308613871 created "2022-11-13" @default.
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- W4308613871 date "2022-11-06" @default.
- W4308613871 modified "2023-09-28" @default.
- W4308613871 title "Maximum size of $C_{leq k}$-free strong digraphs with out-degree at least two" @default.
- W4308613871 doi "https://doi.org/10.48550/arxiv.2211.03129" @default.
- W4308613871 hasPublicationYear "2022" @default.
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