On the complexity of 4-coloring graphs without long induced paths

We show that deciding if a graph without induced paths on nine vertices can be colored with 4 colors is an NP-complete problem, improving a previous NP-completeness result proved by Woeginger and Sgall in 2001. The complexity of 4-coloring graphs without induced paths on five vertices remains open. We show that deciding if a graph without induced paths or cycles on five vertices can be colored with 4 colors can be done in polynomial time. A step in our algorithm uses the well-known and deep fact due to Grötschel, Lovász and Schrijver stating that perfect graphs can be optimally colored in polynomial time.