Coloring (P 5, kite)-free graphs with small cliques

Let P n and K n denote the induced path and complete graph on n vertices, respectively. The kite is the graph obtained from a P 4 by adding a vertex and making it adjacent to all vertices in the P 4 except one vertex with degree 1. A graph is (P 5, kite)-free if it has no induced subgraph isomorphic to a P 5 or a kite. For a graph G, the chromatic number of G (denoted by χ ( G )) is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices receive the same color, and the clique number of G is the size of a largest clique in G. Here, we are interested in coloring the class of (P 5, kite)-free graphs with small clique number. It is known that every (P 5,  kite, K 3)-free graph G satisfies χ ( G ) ≤ 3, every (P 5,  kite, K 4)-free graph G satisfies χ ( G ) ≤ 4, and that every (P 5,  kite, K 5)-free graph G satisfies χ ( G ) ≤ 6. In this paper, we show the following:We also give examples to show that the above bounds are tight. Based on these partial results and some other examples, we conjecture that every (P 5, kite)-free graph G satisfies χ ( G ) ≤ ⌊ 3 ω ( G ) 2 ⌋, and that the bound is tight.