  This article foucuses on $(P_3\cup P_2,K_4)$-free graph. In this paper, we
prove that if G is $(P_3\cup P_2,K_4)$-free, then $\chi(G)\le 7$. We then use
our result to obtain the upper bound of order and chromatic number of
$(4K_1,\overline{P_3\cup P_2},K_{\omega})$-free graph .
