关于有向图中广义拟核的三个结果
Three Results on Generalized Quasikernels in Digraphs
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中文总结 AI 辅助
本文研究有向图中广义拟核，通过证明无向\(4 -\)圈的\(n\)顶点无源二分有向图有特定大小拟核、无\((r - 1)\)源集的有向图含\(r\)个不相交\((3r - 2)\)核，以及单圈有向图中存在特定的两个\(q -\)核，证实猜想并回答问题。
中文摘要 AI 辅助
有向图\(D\)的\(q -\)核是一个独立集\(Q\subseteq D\)，使得\(D\)的每个顶点都可通过长度至多为\(q\)的有向路径从\(Q\)到达，这是核和拟核的自然推广。本文建立了关于广义拟核的三个结果。一是证明无向\(4 -\)圈的\(n\)顶点无源二分有向图有大小至多为\(17n/35\)的拟核；二是表明无\((r - 1)\)源集的每个有向图包含\(r\)个两两不相交的\((3r - 2)\)核；三是考虑长度为\(2\ell\)且二分\(U\cup V\)的单圈有向图，证明对于每个奇数\(q\geq3\)，存在两个\(q -\)核\(Q_U\subseteq U\)和\(Q_V\subseteq V\)使得\(|Q_U| + |Q_V|\leq2\cdot\frac{\lceil\ell/(q + 1)\rceil}{\ell}|V(D)|\)。这些结果证实了两个猜想并回答了一个问题。
英文摘要
A $q$-kernel of a digraph $D$ is an independent set $Q\subseteq D$ such that every vertex of $D$ is reachable from $Q$ by a directed path of length at most $q$, which is a natural generalization of kernels and quasikernels. In this paper, we establish three results on generalized quasikernels. Firstly, we prove that any $n$-vertex source-free bipartite oriented graph with no directed 4-cycles has a quasikernel of size at most $17n/35$. Secondly, we show that every digraph with no $(r-1)$-source set contains $r$ pairwise disjoint $(3r-2)$-kernels, where $r\ge 2$. At last, we consider unicyclic digraph with a directed cycle of length $2\ell$ and bipartition $U\cup V$, and we prove that for every odd integer $q\ge 3$, there exist two $q$-kernels $Q_U\subseteq U$ and $Q_V\subseteq V$ such that \[ |Q_U|+|Q_V| \le 2\cdot \frac{\lceil \ell/(q+1)\rceil}{\ell} |V(D)|. \] These results confirm two conjectures and give an affirmative answer to a question posed by Spiro in European Journal of Combinatorics 133 (2026), 104307.
