Optimal coloring of (P2 + P3, gem)-free graphs

Given a graph G, the parameters $\chi \left(G\right)$ and $\omega \left(G\right)$ respectively denote the chromatic number and the clique number of G. A function $f:N\to N$ such that $f\left(1\right)=1$ and $f\left(x\right)\ge x$, for all $x\in N$ is called a χ-binding function for the given class of graphs $G$ if every $G\in G$ satisfies $\chi \left(G\right)\le f\left(\omega \right(G\left)\right)$, and the smallest χ-binding function $f^{⁎}$ for $G$ is defined as $f^{⁎}\left(x\right):=\max \left\{\chi \right(G\left)\right|G\in G\text{ and }\omega \left(G\right)=x\}$. 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 ($P_2+P_3$, gem)-free graphs, and prove that the function $\varphi :N\to N$ defined by $\varphi \left(1\right)=1$, $\varphi \left(2\right)=4$, $\varphi \left(3\right)=6$ and $\varphi \left(x\right)=\lceil \frac{1}{4}(5x-1)\rceil$, for $x\ge 4$ is the smallest χ-binding function for the class of ($P_2+P_3$, gem)-free graphs. Also we completely characterize the class of ($P_2+P_3$, gem)-free graphs $P$ where every $G\in P$ satisfies $\chi \left(G\right)=\lceil \frac{1}{4}\left(5\omega \right(G)-1)\rceil$.