In this paper, we study the chromatic number for graphs with forbidden induced subgraphs. We improve the existing χ-binding functions for some subclasses of 2K 2 -free graphs, namely {2K 2 , H}-free graphs where H ∈ {K 5 − e, K 2 + P 4 , K 1 + C 4 }. In addition, for p ≥ 3, we find the polynomial χ-binding functions {pK 2 , H}-free graphs where H ∈ {gem, diamond, HV N, K 5 − e, K 2 + P 4 , butterf ly, dart, gem

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Finding families that admit a linear χ-binding function is a problem that has interested researchers for a long time. Recently, the question of finding linear subfamilies of 2K 2 -free graphs has garnered much attention. In this paper, we are interested in finding a linear subfamily of a specific superclass of 2K 2 -free graphs, namely (P 3 ∪ P 2 )-free graphs. We show that the class of {P 3 ∪ P 2 , gem}-free graphs admits f = 2ω as a linear χ-binding function.Furthermore, we give examples to show that the optimal χ-binding function f * ≥ 5ω(G) 4 for the class of {P 3 ∪ P 2 , gem}-free graphs and that the χ-binding function f = 2ω is tight when ω = 2 and 3.

The class of 2K 2 -free graphs have been well studied in various contexts in the past. It is known that the class of {2K 2 , 2K 1 +K p }-free graphs and {2K 2 , (K 1 ∪K 2 )+K p }-free graphs admits a linear χ-binding function. In this paper, we study the classes of (P 3 ∪ P 2 )-free graphs which is a superclass of 2K 2 -free graphs. We show that {P 3 ∪ P 2 , 2K 1 + K p }-free graphs and {P 3 ∪P 2 , (K 1 ∪K 2 )+K p }-free graphs also admits linear χ-binding functions. In addition, we give tight chromatic bounds for {P 3 ∪ P 2 , HV N }-free graphs and {P 3 ∪ P 2 , diamond}-free graphs and it can be seen that the latter is an improvement of the existing bound given by A. P. Bharathi and S. A. Choudum [Colouring of (P 3 ∪ P 2 )-free graphs, Graphs and Combinatorics 34 (2018), 97-107].