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We determined the minimal number of edges of a graph of order $n$ with $\\kappa_{3}= 2$, i.e., for a graph $G$ of order $n$ and size $e(G)$ with $\\kappa_{3}(G)= 2$, we proved that $e(G)\\geq (6/5)n$, and the lower bound is sharp by constructing a class of graphs, only for $n\\equiv 0 \\ (mod \\ 5)$ and $n\\neq 10$. In this paper, we improve the lower bound to $\\lceil(6/5)n\\rceil$. 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