The Chromatic Number of (P₅, HVN)-free Graphs

Let G be a graph. We use χ(G) and ω(G) to denote the chromatic number and clique number of G respectively. A P₅ is a path on 5 vertices, and an HVN is a K₄ together with one more vertex which is adjacent to exactly two vertices of K₄. Combining with some known result, in this paper we show that if G is (P₅, HVN)-free, then χ(G) ≤ max{min{16, ω(G) + 3}, ω(G) + 1}. This upper bound is almost sharp.