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 obtain a kernel of bitsize O(kq−1 log k) for q-Coloring parameterized by Vertex Cover for any q ≥ 3. This size bound is optimal up to ko(1) factors assuming NP 6⊆ coNP/poly, and improves on the previous-best kernel of size O(k). We generalize this result for deciding q-colorability of a graph G, to deciding the existence of a homomorphism from G to an arbitrary fixed graph H. Furthermore, we can replace the parameter vertex cover by the less restrictive parameter twin-cover. We prove that H-Coloring parameterized by Twin-Cover has a kernel of size O(k∆(H) log k). Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP 6⊆ coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size O(n2−ε) for any ε > 0. Previously, such a lower bound was only known for coloring with q ≥ 4 colors.