Kernelization for list $H$-coloring for graphs with small vertex cover

Abstract: For a fixed graph $H$, in the List $H$-Coloring problem, we are given a graph $G$ along with list $L(v) \subseteq V(H)$ for every $v \in V(G)$, and we have to determine if there exists a list homomorphism $\varphi$ from $(G,L)$ to $H$, i.e., an edge preserving mapping $\varphi: V(G)\to V(H)$ that satisfies $\varphi(v)\in L(v)$ for every $v\in V(G)$. Note that if $H$ is the complete graph on $q$ vertices, the problem is equivalent to List $q$-Coloring. We investigate the kernelization properties of List $H$-Coloring parameterized by the vertex cover number of $G$: given an instance $(G,L)$ and a vertex cover of $G$ of size $k$, can we reduce $(G,L)$ to an equivalent instance $(G&#39;,L&#39;)$ of List $H$-Coloring where the size of $G&#39;$ is bounded by a low-degree polynomial $p(k)$ in $k$? This question has been investigated previously by Jansen and Pieterse [Algorithmica 2019], who provided an upper bound, which turns out to be optimal if $H$ is a complete graph, i.e., for List $q$-Coloring. This result was one of the first applications of the method of kernelization via bounded-degree polynomials. We define two new integral graph invariants, $c<sup>*(H)$ and $d<sup>*(H)$, with $$d^*(H) \leq c^*(H) \leq d<sup>*(H)+1$$, and show that for every graph $H$, List $H$-Coloring -- has a kernel with $\mathcal{O}(k<sup>{c<sup>*(H)})$ vertices, -- admits no kernel of size $\mathcal{O}(k<sup>{d<sup>*(H)-\varepsilon})$ for any $\varepsilon &gt; 0$, unless the polynomial hierarchy collapses. -- Furthermore, if $c^*(H) &gt; d<sup>*(H)$, then there is a kernel with $\mathcal{O}(k<sup>{c<sup>*(H)-\varepsilon})$ vertices where $\varepsilon \geq 2<sup>{1-c<sup>*(H)}$. Additionally, we show that for some classes of graphs, including powers of cycles and graphs $H$ where $\Delta(H) \leq c<sup>*(H)$ (which in particular includes cliques), the bound $d<sup>*(H)$ is tight, using the polynomial method. We conjecture that this holds in general.