We investigate the List $H$-Coloring problem, the generalization of graph coloring that asks whether an input graph $G$ admits a homomorphism to the undirected graph $H$ (possibly with loops), such that each vertex $v \in V(G)$ is mapped to a vertex on its list $L(v) \subseteq V(H)$. An important result by Feder, Hell, and Huang [JGT 2003] states that List $H$-Coloring is polynomial-time solvable if $H$ is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an $n$-vertex instance be efficiently reduced to an equivalent instance of bitsize $O(n^{2-\varepsilon})$ for some $\varepsilon > 0$? We prove that if $H$ is not a bi-arc graph, then List $H$-Coloring does not admit such a sparsification algorithm unless $NP \subseteq coNP/poly$. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs $H$ which are not bi-arc graphs.

中文翻译：

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列表 $H$-Coloring 的稀疏化下限

我们研究 List $H$-Coloring 问题，图着色的泛化问题，询问输入图 $G$ 是否承认无向图 $H$（可能带有循环）的同态性，使得每个顶点 $v \in V (G)$ 映射到其列表 $L(v) \subseteq V(H)$ 上的顶点。Feder、Hell 和 Huang [JGT 2003] 的一个重要结果表明，如果 $H$ 是所谓的双弧图，则列表 $H$-Coloring 是多项式时间可解的，否则为 NP-完全图。我们从多项式时间稀疏化的角度研究问题的 NP 完全情况：可以将 $n$-顶点实例有效地减少到位大小 $O(n^{2-\varepsilon})$ 的等效实例一些 $\varepsilon > 0$？我们证明，如果$H$ 不是双弧图，则List $H$-Coloring 不承认这种稀疏化算法，除非$NP \subseteq coNP/poly$。