On the chromatic number of $$P_5$$ -free graphs with no large intersecting cliques

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 \(\chi (G)\le 9\omega (G)\).