Kernelization for $H$-Coloring

Abstract

For a fixed graph $H$, the $H$-Coloring problem asks whether a given graph admits an edge-preserving function from its vertex set to that of $H$. A seminal theorem of Hell and Ne\v{s}et\v{r}il asserts that the $H$-Coloring problem is NP-hard whenever $H$ is loopless and non-bipartite. A result of Jansen and Pieterse implies that for every graph $H$, the $H$-Coloring problem parameterized by the vertex cover number $k$ admits a kernel with $O(k^{\Delta(H)})$ vertices and bit-size bounded by $O(k^{\Delta(H)} \cdot \log k)$, where $\Delta(H)$ denotes the maximum degree in $H$. For the case where $H$ is a complete graph on at least three vertices, this kernel size nearly matches conditional lower bounds established by Jansen and Kratsch and by Jansen and Pieterse. This paper presents new upper and lower bounds on the kernel size of $H$-Coloring problems parameterized by the vertex cover number. The upper bounds arise from two kernelization algorithms. The first is purely combinatorial, and its size is governed by a structural quantity of the graph $H$, called the non-adjacency witness number. As applications, we obtain kernels whose size is bounded by a fixed polynomial for natural classes of graphs $H$ with unbounded maximum degree. More strikingly, we show that for almost every graph $H$, the degree of the polynomial that bounds the size of our combinatorial kernel grows only logarithmically in $\Delta(H)$. Our second kernel leverages linear-algebraic tools and involves the notion of faithful independent representations of graphs. It strengthens the general bound from prior work and, among other applications, yields near-optimal kernels for problems concerning the dimension of orthogonal graph representations over finite fields. We complement these results with conditional lower bounds, thereby nearly settling the kernel complexity of the problem for various target graphs $H$.