A near-optimal kernel for a coloring problem
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≥3, the q− Coloring problem parameterized by the vertex cover number...
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| Published in: | Discrete Applied Mathematics Vol. 377; pp. 66 - 73 |
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31.12.2025
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| Abstract | 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≥3, the q− Coloring problem parameterized by the vertex cover number k admits a kernel of bit-size O˜(kq−1), but admits no kernel of bit-size O(kq−1−ɛ) for ɛ>0 unless NP⊆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≥3, the problem admits a kernel of bit-size O˜(k2q−2), but admits no kernel of bit-size O(k2q−3−ɛ) for ɛ>0 unless NP⊆coNP/poly. He further proved that for q∈{3,4} the problem admits a near-optimal kernel of bit-size O˜(k2q−3) and asked whether such a kernel is achievable for all integers q≥3. In this short paper, we settle this question in the affirmative. |
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| AbstractList | 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≥3, the q− Coloring problem parameterized by the vertex cover number k admits a kernel of bit-size O˜(kq−1), but admits no kernel of bit-size O(kq−1−ɛ) for ɛ>0 unless NP⊆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≥3, the problem admits a kernel of bit-size O˜(k2q−2), but admits no kernel of bit-size O(k2q−3−ɛ) for ɛ>0 unless NP⊆coNP/poly. He further proved that for q∈{3,4} the problem admits a near-optimal kernel of bit-size O˜(k2q−3) and asked whether such a kernel is achievable for all integers q≥3. In this short paper, we settle this question in the affirmative. |
| Author | Haviv, Ishay Rabinovich, Dror |
| Author_xml | – sequence: 1 givenname: Ishay orcidid: 0000-0002-2903-076X surname: Haviv fullname: Haviv, Ishay email: ishayhav@mta.ac.il – sequence: 2 givenname: Dror surname: Rabinovich fullname: Rabinovich, Dror |
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| Cites_doi | 10.1145/3349618 10.1016/j.ic.2013.08.005 10.1016/j.tcs.2010.10.043 10.1016/0304-3975(83)90020-8 10.1145/3624704 10.1007/978-3-540-30559-0_22 10.1007/s00453-019-00660-y 10.1007/s00453-019-00578-5 10.1016/S0166-218X(02)00242-1 |
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| SubjectTerms | Graph coloring Kernelization Linear algebra Parameterized complexity |
| Title | A near-optimal kernel for a coloring problem |
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