A near-optimal kernel for a coloring problem

For a fixed integer $q$, the $q\text{− Coloring}$ 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 $q\text{− Coloring}$ problem parameterized by the vertex cover number $k$ admits a kernel of bit-size $\tilde{O}\left(k^{q-1}\right)$, but admits no kernel of bit-size $O\left(k^{q-1-\varepsilon }\right)$ for $\varepsilon >0$ unless $\mathrm{NP}\subseteq \mathrm{coNP/poly}$ (Jansen and Kratsch, 2013; Jansen and Pieterse, 2019). In 2020, Schalken proposed the question of the kernelizability of the $q\text{− Coloring}$ 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 $\tilde{O}\left(k^{2q-2}\right)$, but admits no kernel of bit-size $O\left(k^{2q-3-\varepsilon }\right)$ for $\varepsilon >0$ unless $\mathrm{NP}\subseteq \mathrm{coNP/poly}$. He further proved that for $q\in \{3,4\}$ the problem admits a near-optimal kernel of bit-size $\tilde{O}\left(k^{2q-3}\right)$ 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.