Optimal data reduction for graph coloring using low-degree polynomials

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we use a recent technique of finding redundant constraints by representing them as lowdegree polynomials, to obtain a kernel of bitsize O(k^q-1 log k) for q-Coloring parameterized by Vertex Cover for any q ≥ 3. This size bound is optimal up to k^o(1) factors assuming NP ⊈ coNP/poly, and improves on the previous-best kernel of size O(k^q). Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP ⊈ coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size O(n^2-ϵ) for any ϵ > 0. Previously, such a lower bound was only known for coloring with q ≥ 4 colors.