摘要：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 â^^ V(G) is mapped to a vertex on its list L(v) âS 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 ð'ª(n^(2-ε)) 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 NP âS 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.
关键词：List H-Coloring; Sparsification; Constraint Satisfaction Problem
