## Abstract

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

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This research was supported by the National Natural Science Foundation of China (No. 12071265), the Shandong Provincial Natural Science Foundation (No. ZR2019MA032).

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Xu, W., Zhang, X. On the chromatic number of \(P_5\)-free graphs with no large intersecting cliques.
*J Comb Optim* **46**, 22 (2023). https://doi.org/10.1007/s10878-023-01088-5

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DOI: https://doi.org/10.1007/s10878-023-01088-5