A Near-Optimal Kernel for a Coloring Problem
Ishay Haviv, Dror Rabinovich
TL;DR
The paper resolves the kernelization question for the $q$-Coloring problem on ${\textsc{Independent Edge}}+k\mathrm{v}$ graphs for all $q\ge 3$ by achieving a near-optimal kernel of size $O(k^{2q-3})$ vertices and bit-size $O(k^{2q-3}\log k)$. It introduces a sparsification framework that encodes colors as vectors over a field via a $q$-palette and enforces coloring constraints with low-degree polynomials of degrees $q-1$, $q-2$, and $2q-3$. The key technical contributions are the constructive existence of $q$-palettes, the corresponding polynomial encodings, and their integration into Schalken's kernelization approach to bound instance size without loss of colorability. This advances kernelization for coloring problems under distance-to-graph-family parameters and suggests broader applicability of polynomial sparsification to related combinatorial problems.
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 $q \geq 3$, 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.