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 $$H_1,H_2,\ldots $$H 1 , H 2 , … ; the graphs in the class are called $$(H_1,H_2,\ldots )$$( 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 $$2P_4$$2 P 4 and $$P_8$$P 8 . For $$P_8$$P 8 -free graphs, some partial results are known, but to the best of our knowledge, $$2P_4$$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 $$(2P_4,C_5)$$( 2 P 4 , C 5 ) -free graphs.