Wheel-free graphs with no induced five-vertex path

04/03/2020

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by   Arnab Char, et al.

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A 4-wheel is the graph consisting of a chordless cycle on four vertices C_4 plus an additional vertex adjacent to all the vertices of the C_4. In this paper, we explore the structure of (P_5,4-wheel)-free graphs, and show that every such graph G is either perfect, or a quasi-line graph, or has a clique cutset, or G belongs to some well-defined special classes of graphs. This result enables us to show that every (P_5,4-wheel)-free graph G satisfies χ(G)≤3/2ω(G). Moreover, this bound is asymptotically tight. That is, there is a class of (P_5,4-wheel)-free graphs H such that every graph H∈ H satisfies χ(H)≥10/7ω(H).