Updating the complexity status of coloring graphs without a fixed induced linear forest

Abstract

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 ℓ - Coloring problem is the problem to decide whether a graph can be colored with at most ℓ colors such that adjacent vertices receive different colors. We show that 4- Coloring is NP-complete for P 8 -free graphs. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4- Coloring is NP-complete for P 9 -free graphs, and a result of Woeginger and Sgall, who showed that 5- Coloring is NP-complete for P 8 -free graphs. Additionally, we prove that the precoloring extension version of 4 - Coloring is NP-complete for P 7 -free graphs, but that the precoloring extension version of 3 - Coloring can be solved in polynomial time for ( P 2 + P 4 ) -free graphs, a subclass of P 7 -free graphs. Here P 2 + P 4 denotes the disjoint union of a P 2 and a P 4 . We denote the disjoint union of s copies of a P 3 by s P 3 and involve Ramsey numbers to prove that the precoloring extension version of 3 - Coloring can be solved in polynomial time for s P 3 -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 for H -free graphs when H is a fixed graph on at most 6 vertices: the problem is polynomial-time solvable if H is a linear forest; otherwise it is NP-complete.