Rainbow disjoint union of clique and matching in edge-colored complete graph

Given an edge-coloring of a graph G, G is said to be rainbow if any two edges of G receive different colors. The anti-Ramsey number AR(G, H) is defined to be the maximum integer k such that there exists a k-edgecoloring of G avoiding rainbow copies of H. The anti-Ramsey number for graphs, especially matchings, have been studied in several graph classes. Gilboa and Roditty focused on the anti-Ramsey number of graphs with small components, especially including a matching. In this paper, we continue the work in this direct and determine the exact value of the anti-Ramsey number of K4 boolean OR tP2 in complete graphs. Also, we improve the bound and obtain the exact value of AR(Kn, C3 boolean OR tP2) for all n >= 2t + 3.