A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph P k denotes a path on k vertices. The -Coloring problem is the problem to decide whether a graph can be colored with at most colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P8-free graphs. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4-Coloring is NP-complete for P9-free graphs, and a result of Woeginger and Sgall, who showed that 5-Coloring is NPcomplete for P8-free graphs. Additionally, we prove that the precoloring extension version of 4-Coloring is NP-complete for P7-free graphs, but that the precoloring extension version of 3-Coloring can be solved in polynomial time for (P2 + P4)-free graphs, a subclass of P7-free graphs. Here P2 + P4 denotes the disjoint union of a P2 and a P4. We denote the disjoint union of s copies of a P3 by sP3 and involve Ramsey numbers to prove that the precoloring extension version of 3-Coloring can be solved in polynomial time for sP3-free graphs for any fixed s. Combining our last two results with known results yields a complete complexity classification of (precoloring extension of) 3-Coloring
Later, Broersma et al. REF extended this result by showing that 3-PRECOLORING EXTENSION can be solved in polynomial time for H -free graphs if H is a linear forest on at most 6 vertices.
Updating the complexity status of coloring graphs without a fixed induced linear forest
Theor. Comput. Sci.
Theor. Comput. Sci.
Mathematics Computer Science