On the chromatic number of some P5-free graphs

Abstract

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