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

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