  An edge-colored graph is called rainbow if all the colors on its edges are
distinct. Given a positive integer n and a graph G, the anti-Ramsey number
ar(n,G) is the maximum number of colors in an edge-coloring of K_{n} with no
rainbow copy of G. Denote by kC_{3} the union of k vertex-disjoint copies of
C_{3}. In this paper, we determine the anti-Ramsey number ar(n,kC_{3}) for n=3k
and n\geq2k^{2}-k+2, respectively. When 3k\leq n\leq 2k^{2}-k+2, we give lower
and upper bounds for ar(n, kC_{3}).
