AI 中文总结
本文针对Spiro提出的有向图最小$q$-核的三个问题，对其中一个给出肯定回答、两个给出否定回答，还在二分有向图背景下研究了后两个问题并得出相关结论。
AI 中文摘要
我们解决了Spiro[《欧洲组合学杂志》，133：论文编号104307，2026]提出的关于有向图中最小$q$-核的三个问题。这三个问题均围绕有向图中小$q$-核的存在性展开，其中“小”的衡量标准各不相同。我们对其中一个问题给出了肯定回答，对另外两个给出了否定回答。我们进一步在二分有向图的背景下研究后两个问题，特别地，我们证明在二分设定下，其中一个问题的答案为当且仅当$q$为奇数时成立；对于另一个问题，我们证明当有向图为二分图且$q$为奇数时答案为正（而有向图为二分图且$q$为偶数的情况仍未解决）。
英文摘要
We address three questions of Spiro [Europ. J. Combin., 133:Paper No. 104307, 2026] on smallest $q$-kernels in digraphs. All three questions are on the existence of small $q$-kernels in digraphs, where 'smallness' is measured in different ways. We answer one of these questions in the affirmative, and the other two in the negative. We further study the latter two questions in the context of bipartite digraphs. In particular, we show that in the bipartite setting, the answer to one of the two questions is affirmative if and only if $q$ is odd. For the other question, we show that the answer is positive when the digraph is bipartite and $q$ is odd (the case when the digraph is bipartite and $q$ is even remains open).
Comments10 pages, 3 figures
