Improved bounds on the chromatic number of (P5, flag)-free graphs

Abstract

Given a positive integer t, let Pt and Kt respectively denote the chordless path and the complete graph on t vertices. For a graph G, let χ(G) and ω(G) respectively denote the chromatic number and clique number of G. It is known that every (P5,K4)-free graph G satisfies χ(G)≤5, and the bound is tight. A flag is the graph obtained from a K4 by attaching a pendent vertex. Clearly, the class of flag-free graphs generalizes the class of K4-free graphs. In this paper, we show the following:•Every (P5,flag,K5)-free graph G that contains a K4 satisfies χ(G)≤8.•Every (P5,flag,K6)-free graph G satisfies χ(G)≤8.•Every (P5,flag,K7)-free graph G satisfies χ(G)≤9. We also give examples to show that the given bounds are tight. Further, we show that every (P5, flag)-free graph G with ω(G)≥4 satisfies χ(G)≤max⁡{8,2ω(G)−3}, and the bound is tight for ω(G)∈{4,5,6}. We note that our bound is an improvement over that given in Dong et al. [3,4].