Closing complexity gaps for coloring problems on H-free graphs.

If a graph G contains no subgraph isomorphic to some graph H , then G is called H -free. A coloring of a graph G=(V,E) is a mapping c:V→{1,2,…} such that no two adjacent vertices have the same color, i.e., c(u)≠c(v) if uv∈E; if |c(V)|⩽k then c is a k -coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k . The Precoloring Extension problem is to decide whether a partial k -coloring of a graph can be extended to a k -coloring of the whole graph for some integer k . The List Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u). By imposing an upper bound ℓ on the size of each L(u) we obtain the ℓ -List Coloring problem. We first classify the Precoloring Extension problem and the ℓ -List Coloring problem for H -free graphs. We then show that 3-List Coloring is NP-complete for n -vertex graphs of minimum degree n−2, i.e., for complete graphs minus a matching, whereas List Coloring is fixed-parameter tractable for this graph class when parameterized by the number of vertices of degree n−2. Finally, for a fixed integer k>0, the Listk -Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u) that must be a subset of {1,…,k}. We show that List 4-Coloring is NP-complete for P6-free graphs, where P6 is the path on six vertices. This completes the classification of Listk -Coloring for P6-free graphs