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 mK₃ 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. European J. Combin. 36 (2014) found that a graph must contain a rainbow triangle if e(G)+c(G) ≥ n2+ n. L. Li. and X. Li. Discrete Applied Mathematics 318 (2022) conjectured a lower bound on $e(G)+c(G)$ such that G must contain a proper mK₃. 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 Kₖ subgraphs in general host graphs, and a sharp result on the existence of proper mK₃ in complete graphs.