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 \(\mathcal {O}(n^{2-\varepsilon }) \) for some ε > 0? We prove that if H is not a bi-arc graph, then List H -Coloring does not admit such a sparsification algorithm unless \({\mathsf {NP \subseteq coNP/poly}} \) . Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi- graphs.