Note on vertex disjoint rainbow triangles in edge-colored graphs

Given an edge-colored graph $G$, we denote the number of colors as $c(G)$, and the number of edges as $e(G)$. An edge-colored graph is rainbow if no two edges share the same color. A proper $m{K}_3$ is a vertex disjoint union of $m$ rainbow triangles. Rainbow problems have been studied extensively in the context of anti-Ramsey theory, and more recently, in the context of Turán problems. B. Li. et al. $\mathit{\text{European}}\mathit{\text{J.}}\mathit{\text{Combin.}}\mathit{\text{36}}\mathit{\text{(2014)}}$ found that a graph must contain a rainbow triangle if $e(G)+c(G)\ge \left(\genfrac{}{}{0px}{}{n}{2}\right)+n$. L. Li. and X. Li. $\mathit{\text{Discrete}}\mathit{\text{Applied}}\mathit{\text{Mathematics}}\mathit{\text{318}}\mathit{\text{(2022)}}$ conjectured a lower bound on $e(G)+c(G)$ such that $G$ must contain a proper $m{K}_3$. In this paper, we provide a construction that disproves the conjecture. We also introduce a result that guarantees the existence of $m$ vertex disjoint rainbow $K_k$ subgraphs in general host graphs, and a sharp result on the existence of proper $m{K}_3$ in complete graphs.