On the chromatic number of some $P_5$-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 P t and C t to denote a path and a cycle on t vertices, respectively. For two disjoint graphs F 1 and F 2 , we use F 1 ∪ F 2 to denote the graph with vertex set V (F 1 ) ∪ V (F 2 ) and edge set E(F 1 ) ∪ E(F 2 ), and use F 1 + F 2 to denote the graph with vertex setIn this paper, we prove that (i) (P 5 , C 5 , K 2,3 )-free graphs are perfectly divisible, (ii)free, and (iv) χ(G) ≤ 3ω(G)+11 if G is (P 5 , K 1 +(K 1 ∪K 3 ))-free.

$χ$-binding function for a superclass of $2K_2$-free graphs

“…Figure 1: Some special graphs {P 5 , butterf ly}-free graphs which was recently improved to 3 2 (ω 2 − ω) by Wei Dong et al in [6]. Finally, it remains to study the χ-binding function for {butterf ly, hammer}-free graphs.…”