Coloring of some (P2 ∪ P4)-free graphs

A diamond (resp. gem) is a graph consisting of a [Formula: see text] (resp. [Formula: see text]) and a new vertex adjacent to all vertices of the [Formula: see text] (resp. [Formula: see text]), and a butterfly is a graph consisting of two triangles that share one vertex. In this paper, we show that [Formula: see text] if [Formula: see text] is a ([Formula: see text], gem)-free graph, and for a ([Formula: see text], butterfly)-free graph [Formula: see text], [Formula: see text]. We also study the class of ([Formula: see text], diamond)-free graphs, and prove that for such a graph [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text] if [Formula: see text]. Moreover, we prove that [Formula: see text] is perfect if [Formula: see text] is ([Formula: see text], diamond, [Formula: see text])-free with [Formula: see text]. These results generalize several theorems by Francis et al.