The chromatic number of {P5,K4}{P5,K4}-free graphs

Gyárfás conjectured that for any tree TT every TT-free graph GG with maximum clique size ω(G)ω(G) is fT(ω(G))fT(ω(G))-colorable, for some function fTfT that depends only on TT and ω(G)ω(G). Moreover, he proved the conjecture when TT is the path PkPk on kk vertices. In the case of P5P5, the best values or bounds known so far are fP5(2)=3fP5(2)=3 and fP5(q)≤3q−1fP5(q)≤3q−1. We prove here that fP5(3)=5fP5(3)=5.