Abstrakt
The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs H_1, H_2, ...; the graphs in the class are called (H_1, H_2, ...)-free.
The complexity of 3-coloring is far from being understood, even for classes defined by a few small forbidden induced subgraphs. For H-free graphs, the complexity is settled for any H on up to seven vertices.
There are only two unsolved cases on eight vertices, namely 2P_4 and P_8. For P_8-free graphs, some partial results are known, but to the best of our knowledge, 2P_4-free graphs have not been explored yet.
In this paper, we show that the 3-coloring problem is polynomial-time solvable on (2P_4, C_5)-free graphs.