Colouring of $$(P_3 \cup P_2)$$-free graphs

The class of \(2K_{2}\)-free graphs and its various subclasses have been studied in a variety of contexts. In this paper, we are concerned with the colouring of \((P_{3}\cup P_{2})\)-free graphs, a super class of \(2K_{2}\)-free graphs. We derive a \(O(\omega ^{3})\) upper bound for the chromatic number of \((P_{3} \cup P_{2})\)-free graphs, and sharper bounds for \((P_{3} \cup P_{2}\), diamond)-free graphs and for \((2K_{2},\) diamond)-free graphs, where \(\omega \) denotes the clique number. The last two classes are perfect if \(\omega \ge 5\) and \(\ge 4\) respectively.