A Near-Optimal Kernel for a Coloring Problem

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 $q\ge 3$, the $q$-Coloring problem parameterized by the vertex cover number $k$ admits a kernel of bit-size $\tilde{O}(k^{q-1})$, but admits no kernel of bit-size $O(k^{q-1-\epsilon })$ for $\epsilon >0$ unless $\mathsf{NP}\subseteq coNP/poly$ (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 $q\ge 3$, the problem admits a kernel of bit-size $\tilde{O}(k^{2q-2})$, but admits no kernel of bit-size $O(k^{2q-3-\epsilon })$ for $\epsilon >0$ unless $\mathsf{NP}\subseteq coNP/poly$. He further proved that for $q\in \{3,4\}$ the problem admits a near-optimal kernel of bit-size $\tilde{O}(k^{2q-3})$ and asked whether such a kernel is achievable for all integers $q\ge 3$. In this short paper, we settle this question in the affirmative.

12 pages