A near-optimal kernel for a coloring problem - Haviv, Ishay; Rabinovich, Dror (Discrete Applied Mathematics, 2025)
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.