An optimal chromatic bound for (P₂+P₃, gem)-free graphs

Given a graph G, the parameters χ(G) and ω(G) respectively denote the chromatic number and the clique number of G. A function f : N → N such that $f(1) = 1$ and f(x) ≥ x, for all x ∈ N is called a χ-binding function for the given class of graphs G if every G ∈ G satisfies χ(G) ≤ f(ω(G)), and the smallest χ-binding function f^* for G is defined as f^*(x) := max\χ(G) G∈ G and ω(G)=x\. In general, the problem of obtaining the smallest χ-binding function for the given class of graphs seems to be extremely hard, and only a few classes of graphs are studied in this direction. In this paper, we study the class of (P₂+ P₃, gem)-free graphs, and prove that the function φ:N→ N defined by φ(1)=1, φ(2)=4, φ(3)=6 and φ(x)=1/4(5x-1), for x≥ 4 is the smallest χ-binding function for the class of (P₂+ P₃, gem)-free graphs.