On the chromatic number of (P_{5},windmill)-free graphs

Abstract

In this paper we study the chromatic number of \((P_5, windmill)\)-free graphs. For integers \(r,p\geq 2\) the windmill graph \(W_{r+1}^p=K_1 \vee pK_r\) is the graph obtained by joining a single vertex (the center) to the vertices of \(p\) disjoint copies of a complete graph \(K_r\). Our main result is that every \((P_5, windmill)\)-free graph \(G\) admits a polynomial \(\chi\)-binding function. Moreover, we will present polynomial \(\chi\)-binding functions for several other subclasses of \(P_5\)-free graphs.

Keywords

• vertex colouring
• perfect graphs
• \(\chi\)-binding function
• forbidden induced subgraph