  Let $P_n$ and $K_n$ denote the induced path and complete graph on $n$
vertices, respectively. The {\em 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 $\chi(G)$) 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 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
$\chi(G)\leq 3$, every ($P_5$,~kite, $K_4$)-free graph $G$ satisfies
$\chi(G)\leq 4$, and that every ($P_5$,~kite, $K_5$)-free graph $G$ satisfies
$\chi(G)\leq 6$. In this paper, we showed the following:
  $\bullet$ Every ($P_5$, kite, $K_6$)-free graph $G$ satisfies $\chi(G)\leq
7$.
  $\bullet$ Every ($P_5$, kite, $K_7$)-free graph $G$ satisfies $\chi(G)\leq
9$.
  We also give examples to show that the above bounds are tight.
