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) 6= 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 ` on the size of each L(u) we obtain the `-List Coloring problem. We first classify the Precoloring Extension problem and the `-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 List k-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 NPcomplete for P6-free graphs, where P6 is the path on six vertices. This completes the classification of List k-Coloring for P6-free graphs.