拟核猜想的一个可变版本
A variable version of the quasi-kernel conjecture
AI总结：
本文研究有向图拟核的大、小 QK 猜想之间的关系，证明二者在尖锐形式下的等价性及带较弱常数的蕴含关系，并对核完美数为 $k$ 的有向图给出大小至多 $(1-1/k)n$ 的拟核。
摘要：
有向图 $D$ 的拟核是一个独立集 $Q$，使得每个顶点都能在至多两步内到达 $Q$。由 P.L. Erdős 和 Székely 提出、有着 48 年历史的一个猜想被称为小 QK 猜想，它指出每个无汇点有向图都包含一个大小至多为 $n/2$ 的拟核。最近，Spiro 提出了大 QK 猜想，即每个无汇点有向图都包含一个拟核 $Q$，满足 $|N^-[Q]|\geq n/2$，并证明它可由小 QK 猜想推出。本文证明大 QK 猜想蕴含一个常数较弱的小 QK 猜想。我们还证明大 QK 猜想等价于它的一个尖锐版本，从而肯定地回答了 Spiro 的一个问题。我们表述了这些猜想的可变版本，它们在一般情形下仍未解决。已知具有大小为 $(1-α)n$ 或更小拟核的有向图并不多。我们对二色数有界的有向图证明了这一点，具体是通过证明更强的命题：每个无汇点有向图都包含一个大小至多为 $(1-1/k)n$ 的拟核，其中 $k$ 是该有向图的核完美数。
英文摘要：
A quasi-kernel of a digraph $D$ is an independent set $Q$ such that every vertex can reach $Q$ in at most two steps. A 48-year conjecture made by P.L. Erdős and Székely, denoted the small QK conjecture, says that every sink-free digraph contains a quasi-kernel of size at most $n/2$. Recently, Spiro posed the large QK conjecture, that every sink-free digraph contains a quasi-kernel $Q$ such that $|N^-[Q]|\geq n/2$, and showed that it follows from the small QK conjecture. In this paper, we establish that the large QK conjecture implies the small QK conjecture with a weaker constant. We also show that the large QK conjecture is equivalent to a sharp version of it, answering affirmatively a question of Spiro. We formulate variable versions of these conjectures, which are still open in general. Not many digraphs are known to have quasi-kernels of size $(1-α)n$ or less. We show this for digraphs with bounded dichromatic number, by proving the stronger statement that every sink-free digraph contains a quasi-kernel of size at most $(1-1/k)n$, where $k$ is the digraph's kernel-perfect number.
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