Sparsification Lower Bounds for List H-Coloring

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. V (G) is mapped to a vertex on its list L(v). 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-epsilon)) for some epsilon > 0? We prove that ifH is not a bi-arc graph, then ListH-Coloring does not admit such a sparsification algorithm unless NP. coNP/poly. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-graphs.