On 3-Coloring of ( 2P_4,C_5 )-Free Graphs

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.