TheChromaticNumberof（P5,HVN）-freeGraphs

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.