有向图中3q-有向环的指定顶点半度阈值
The Prescribed-Vertex Semidegree Threshold for Directed $3q$-Cycles in Oriented Graphs
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中文总结 AI 辅助
该研究确定有向图中3q-有向环的指定顶点半度阈值，填补相关定理的缺口，优化了一般指定顶点定理的阶数假设。
中文摘要 AI 辅助
对于每个q≥2，我们证明：每个顶点数n≥45q−8的有向图G，若其最小半度δ⁰(G)≥⌈n/3⌉，则G中每个顶点都包含长度为3q的有向环。该半度阈值是紧的。当3整除n时，这填补了Kelly、Kühn和Osthus的指定顶点定理留下的1个单位的缺口。我们还证明：若有向图H的阶数为N，最小半度d≥3且7d≥2N+3，则每对不同顶点都由长度为3、4或5的路径连接。常数+3是最优的。由此，Kelly、Kühn和Osthus的一般指定顶点定理中阶数假设n≥10¹⁰ℓ可替换为n≥15ℓ−60（ℓ≥7）。
英文摘要
For every $q\ge2$, we prove that every oriented graph $G$ on $n\ge45q-8$ vertices whose minimum semidegree satisfies \[ δ^0(G)\ge \left\lceil\frac n3\right\rceil \] contains a directed cycle of length $3q$ through every vertex. The semidegree bound is sharp. This closes the one-unit gap left by the prescribed-vertex theorem of Kelly, Kühn and Osthus when $3\mid n$. We also prove that if an oriented graph $H$ has order $N$, minimum semidegree $d\ge3$, and $7d\ge2N+3$, then every ordered pair of distinct vertices is joined by a path of length three, four, or five. The constant $+3$ is best possible. As a consequence, the order hypothesis $n\ge10^{10}\ell$ in the general prescribed-vertex theorem of Kelly, Kühn and Osthus can be replaced by $n\ge15\ell-60$ for $\ell\ge7$.
