On 3-Coloring of ( 2 P 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 2 P 4 and P 8 . For P 8 -free graphs, some partial results are known, but to the best of our knowledge, 2 P 4 -free graphs have not been explored yet. In this paper, we show that the 3-coloring problem is polynomial-time solvable on ( 2 P 4 , C 5 ) -free graphs.