Colouring of $$(P_3 \cup P_2)$$(P3∪P2)-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.