Sparsification Lower Bounds for List H -Coloring

Abstract

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  \(\mathcal {O} (n^{2-\varepsilon })\) ( n 2-ɛ ) for some ɛ > 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.