Note on vertex disjoint rainbow triangles in edge-colored graphs

Abstract: 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 $mK_3$ 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 Tur\'{a}n problems. B. Li. et al. \textit{European J. Combin. 36 (2014)} found that a graph must contain a rainbow triangle if $e(G)+c(G) \geq \binom{n}{2}+ n$. L. Li. and X. Li. \textit{Discrete Applied Mathematics 318 (2022)} conjectured a lower bound on $e(G)+c(G)$ such that $G$ must contain a proper $mK_3$. 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 $mK_3$ in complete graphs.