A variable version of the quasi-kernel conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910484597309440 |
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| author | Ai, Jiangdong Liu, Xiangzhou Peng, Fei |
| author_facet | Ai, Jiangdong Liu, Xiangzhou Peng, Fei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_04887 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A variable version of the quasi-kernel conjecture Ai, Jiangdong Liu, Xiangzhou Peng, Fei Combinatorics 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. |
| title | A variable version of the quasi-kernel conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.04887 |