Coloring Graphs Characterized by a Forbidden Subgraph

The Coloring problem is to test whether a given graph can be colored with at most k colors for some given k, such that no two adjacent vertices receive the same color. The complexity of this problem on graphs that do not contain some graph H as an induced subgraph is known for each fixed graph H. A natural variant is to forbid a graph H only as a subgraph. We call such graphs strongly H-free and initiate a complexity classification of Coloring for strongly H-free graphs. We show that Coloring is NP-complete for strongly H-free graphs, even for k = 3, when H contains a cycle, has maximum degree at least five, or contains a connected component with two vertices of degree four. We also give three conditions on a forest H of maximum degree at most four and with at most one vertex of degree four in each of its connected components, such that Coloring is NP-complete for strongly H-free graphs even for k = 3. Finally, we classify the computational complexity of Coloring on strongly H-free graphs for all fixed graphs H up to seven vertices. In particular, we show that Coloring is polynomial-time solvable when H is a forest that has at most seven vertices and maximum degree at most four.