Abstract
The class of 2 K 2 -free graphs has been well studied in the past. In the paper “On the chromatic number of 2 K 2 -free graphs, Discrete Applied Mathematics, 253 (2019), 14–24”, it was shown that the class of { 2 K 2 , 2 K 1 + K p } -free graphs and { 2 K 2 , ( K 1 ∪ K 2 ) + K p } -free graphs admit a linear χ-binding function. In this paper, we study some subclasses of ( P 3 ∪ P 2 ) -free graphs which is a superclass of 2 K 2 -free graphs. We show that { P 3 ∪ P 2 , 2 K 1 + K p } -free graphs and { P 3 ∪ P 2 , ( K 1 ∪ K 2 ) + K p } -free graphs also admit linear χ-binding functions. In addition, we give a tight χ-binding function for { P 3 ∪ P 2 , HVN } -free graphs and improve the χ-bound for { P 3 ∪ P 2 , diamond } -free graphs with ω = 4.
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