A Near-Optimal Kernel for a Coloring Problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908322704130048 |
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| author | Haviv, Ishay Rabinovich, Dror |
| author_facet | Haviv, Ishay Rabinovich, Dror |
| contents | For a fixed integer $q$, the $q$-Coloring problem asks to decide if a given graph has a vertex coloring with $q$ colors such that no two adjacent vertices receive the same color. In a series of papers, it has been shown that for every $q \geq 3$, the $q$-Coloring problem parameterized by the vertex cover number $k$ admits a kernel of bit-size $\widetilde{O}(k^{q-1})$, but admits no kernel of bit-size $O(k^{q-1-\varepsilon})$ for $\varepsilon >0$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$ (Jansen and Kratsch, 2013; Jansen and Pieterse, 2019). In 2020, Schalken proposed the question of the kernelizability of the $q$-Coloring problem parameterized by the number $k$ of vertices whose removal results in a disjoint union of edges and isolated vertices. He proved that for every $q \geq 3$, the problem admits a kernel of bit-size $\widetilde{O}(k^{2q-2})$, but admits no kernel of bit-size $O(k^{2q-3-\varepsilon})$ for $\varepsilon >0$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$. He further proved that for $q \in \{3,4\}$ the problem admits a near-optimal kernel of bit-size $\widetilde{O}(k^{2q-3})$ and asked whether such a kernel is achievable for all integers $q \geq 3$. In this short paper, we settle this question in the affirmative. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_12281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Near-Optimal Kernel for a Coloring Problem Haviv, Ishay Rabinovich, Dror Data Structures and Algorithms For a fixed integer $q$, the $q$-Coloring problem asks to decide if a given graph has a vertex coloring with $q$ colors such that no two adjacent vertices receive the same color. In a series of papers, it has been shown that for every $q \geq 3$, the $q$-Coloring problem parameterized by the vertex cover number $k$ admits a kernel of bit-size $\widetilde{O}(k^{q-1})$, but admits no kernel of bit-size $O(k^{q-1-\varepsilon})$ for $\varepsilon >0$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$ (Jansen and Kratsch, 2013; Jansen and Pieterse, 2019). In 2020, Schalken proposed the question of the kernelizability of the $q$-Coloring problem parameterized by the number $k$ of vertices whose removal results in a disjoint union of edges and isolated vertices. He proved that for every $q \geq 3$, the problem admits a kernel of bit-size $\widetilde{O}(k^{2q-2})$, but admits no kernel of bit-size $O(k^{2q-3-\varepsilon})$ for $\varepsilon >0$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$. He further proved that for $q \in \{3,4\}$ the problem admits a near-optimal kernel of bit-size $\widetilde{O}(k^{2q-3})$ and asked whether such a kernel is achievable for all integers $q \geq 3$. In this short paper, we settle this question in the affirmative. |
| title | A Near-Optimal Kernel for a Coloring Problem |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.12281 |