Asymptotic value of the minimal size of a graph with rainbow connection number 2

A path in an edge (vertex)-colored graph $G$, where adjacent edges (vertices) may have the same color, is called a rainbow path if no pair of edges (internal vertices) of the path are colored the same. The rainbow (vertex) connection number $rc(G)$ ($rvc(G)$) of $G$ is the minimum integer $i$ for which there exists an $i$-edge (vertex)-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. Denote by ${\mathcal{G}}_d(n)$ (${{\mathcal{G}}^{\mathcal{\prime }}}_d(n)$) the set of all graphs of order $n$ with rainbow (vertex) connection number $d$, and define $e_d(n)=\min{e(G),|, G\in $\mathcal{G}$_d(n)}$($e_d’(n)=\min{e(G),|, G\in ${\mathcal{G}}^{\mathcal{\prime }}$d(n)}$),where$e(G)$denotesthenumberofedgesin$G$. In this paper, we investigate the bounds of $e_2(n)$ and get the exact asymptotic value. i.e., $\displaystyle \lim{n\rightarrow \infty}\frac{e_2(n)}{n\log_2 n}=1$\mathrm{.}Meanwhile,weobtain$e’_d(n)=n-1$for$d\geq 2$,andtheequalityholdsifandonlyif$G$ is such a graph such that deleting all leaves of $G$ results in a tree of order $d$.