A graph G is called \((H_1, H_2)\)-free if G contains no induced subgraph isomorphic to \(H_1\) or \(H_2\). Let \(P_k\) be a path with k vertices and \(C_{s,t,k}\) (\(s\le t\)) be a graph consisting of two intersecting complete graphs \(K_{s+k}\) and \(K_{t+k}\) with exactly k common vertices. In this paper, using an iterative method, we prove that the class of \((P_5,C_{s,t,k})\)-free graphs with clique number \(\omega \) has a polynomial \(\chi \)-binding function \(f(\omega )=c(s,t,k)\omega ^{\max \{s,k\}}\). In particular, we give two improved chromatic bounds: every \((P_5, butterfly)\)-free graph G has \(\chi (G)\le \frac{3}{2}\omega (G)(\omega (G)-1)\); every \((P_5, C_{1,3})\)-free graph G has \(\chi (G)\le 9\omega (G)\).

如果图G不包含与 \ (H_1 \)或\(H_2\)同构的导出子图，则图G称为\((H_1, H_2)\) -free 。设\(P_k\)为具有k 个顶点的路径，\(C_{s,t,k}\) ( \(s\le t\) ) 为由两个相交的完全图组成的图\(K_{s+ k}\)和\(K_{t+k}\)恰好有k 个公共顶点。在本文中，我们使用迭代方法证明了团数为\ (\omega\ )的\((P_5,C_{s,t,k})\)类无图具有多项式\(\chi \) - 绑定函数\(f(\omega )=c(s,t,k)\omega ^{\max \{s,k\}}\)。特别地，我们给出了两个改进的色界：每个\((P_5,蝴蝶)\)无图G都有\(\chi (G)\le \frac{3}{2}\omega (G)(\omega (G)-1)\) ; 每个\((P_5, C_{1,3})\)无图G都有\(\chi (G)\le 9\omega (G)\)。