Coloring graphs with no induced five‐vertex path or gem

For a graph G, let χ(G) and ω(G), respectively, denote the chromatic number and clique number of G. We give an explicit structural description of (P5, gem)‐free graphs, and show that every such graph G satisfies χ(G)≤⌈5ω(G)4⌉. Moreover, this bound is best possible. Here a gem is the graph that consists of an induced four‐vertex path plus a vertex which is adjacent to all the vertices of that path.