Complexity dichotomy for List-5-Coloring with a forbidden induced subgraph [article]

Sepehr Hajebi, Yanjia Li, Sophie Spirkl
2021 arXiv   pre-print
For a positive integer r and graphs G and H, we denote by G+H the disjoint union of G and H, and by rH the union of r mutually disjoint copies of H. Also, we say G is H-free if H is not isomorphic to an induced subgraph of G. We use P_t to denote the path on t vertices. For a fixed positive integer k, the List-k-Coloring Problem is to decide, given a graph G and a list L(v)⊆{1,...,k} of colors assigned to each vertex v of G, whether G admits a proper coloring ϕ with ϕ(v)∈ L(v) for every vertex
more » ... of G, and the k-Coloring Problem is the List-k-Coloring Problem restricted to instances with L(v)={1,..., k} for every vertex v of G. We prove that for every positive integer r, the List-5-Coloring Problem restricted to rP_3-free graphs can be solved in polynomial time. Together with known results, this gives a complete dichotomy for the complexity of the List-5-Coloring Problem restricted to H-free graphs: For every graph H, assuming P≠NP, the List-5-Coloring Problem restricted to H-free graphs can be solved in polynomial time if and only if H is an induced subgraph of either rP_3 or P_5+rP_1 for some positive integer r. As a hardness counterpart, we also show that the k-Coloring Problem restricted to rP_4-free graphs is NP-complete for all k≥ 5 and r≥ 2.
arXiv:2105.01787v2 fatcat:be63sbjvhbfjjdhxt2mvk4el2y