The χ-Boundedness of P2∪P3-Free Graphs - Xiao Wang; Donghan Zhang (Journal of Mathematics)
In the early 1980s, Gyárfás introduced the concept of the χ-bound with χ-binding functions thereby extending the notion of perfectness. There are a number of challenging conjectures about the χ-bound. Let χG, ωG, and ΔG be the chromatic number, clique number, and maximum degree of a graph G, respectively. In this paper, we prove that if G is a triangle-free and P2∪P3-free graph, then χG≤3 unless G is one of eight graphs with ΔG=5 and χG=4, where the eight graphs are extended from the Grötzsch graph as a Mycielskian of a 5-cycle graph. Moreover, we also show that χG≤3ωG if G is a P2∪P3,W4-free graph.