Given a graph $G$, the parameters $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and the clique number of $G$. A function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(1) = 1$ and $f(x) \geq x$, for all $x \in \mathbb{N}$ is called a $\chi$-binding function for the given class of graphs $\cal{G}$ if every $G \in \cal{G}$ satisfies $\chi(G) \leq f(\omega(G))$, and the \emph{smallest $\chi$-binding function} $f^*$ for $\cal{G}$ is defined as $f^*(x) := \max\{\chi(G)\mid G\in {\cal G} \mbox{ and } \omega(G)=x\}$. In general, the problem of obtaining the smallest $\chi$-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 ($P_2+ P_3$, gem)-free graphs, and prove that the function $\phi:\mathbb{N}\rightarrow \mathbb{N}$ defined by $\phi(1)=1$, $\phi(2)=4$, $\phi(3)=6$ and $\phi(x)=\left\lceil\frac{1}{4}(5x-1)\right\rceil$, for $x\geq 4$ is the smallest $\chi$-binding function for the class of ($P_2+ P_3$, gem)-free graphs.

翻译：给定图G，参数χ(G)和ω(G)分别表示G的色数和团数。若函数f: ℕ → ℕ满足f(1)=1且对所有x∈ℕ有f(x)≥x，且对给定图类𝒢中每个图G均有χ(G) ≤ f(ω(G))，则称f为𝒢的χ-约束函数。𝒢的最小χ-约束函数f*定义为f*(x) := max{χ(G) | G∈𝒢 且 ω(G)=x}。一般而言，求解给定图类的最小χ-约束函数是极其困难的问题，目前仅对少数图类开展了相关研究。本文研究(P₂+P₃, gem)-自由图类，证明函数φ:ℕ→ℕ（定义为φ(1)=1，φ(2)=4，φ(3)=6，且对x≥4有φ(x)=⌈(5x-1)/4⌉）是该图类的最小χ-约束函数。