On the chromatic number of some P5-free graphs

Let G be a graph. We say that is perfectly divisible if for each induced subgraph H of G, V(H) can partitioned into A and B such H[A] perfect ω(H[B])<ω(H). use Pt Ct to denote path cycle on t vertices, respectively. For two disjoint graphs F1 F2, we F1∪F2 the graph with vertex set V(F1)∪V(F2) edge E(F1)∪E(F2), F1+F2 E(F1)∪E(F2)∪{xy|x∈V(F1) y∈V(F2)}. In this paper, prove (i) (P5,C5,K2,3)-free are divisible, (ii) χ(G)≤2ω2(G)−ω(G)−3 (P5,K2,3)-free ω(G)≥2, (iii) χ(G)≤32(ω2(G)−ω(G)) (P5,K1+2K2)-free, (iv) χ(G)≤3ω(G)+11 (P5,K1+(K1∪K3))-free.