On 3-Coloring of (2P_4,C_4)-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 1, 2,…; the graphs in the class are called (1, 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 24 and 8. For 8-free graphs, some partial results are known, but to the best of our knowledge, 24-free graphs have not been explored yet. In this paper, we show that the 3-coloring problem is polynomial-time solvable on (24,5)-free graphs.