We call a subgraph of an edge-colored graph rainbow subgraph, if all of its edges have different colors. The anti-Ramsey number of a graph G in a complete graph K n , denoted by ar(K n , G), is the maximum number of colors in an edge-coloring of K n with no rainbow subgraph copy of G. In this paper, we determine the exact value of the anti-Ramsey number for star forests and the approximate value of the anti-Ramsey number for linear forests. Furthermore, we compute the exact value of ar(K n , 2P 4 ) for n ≥ 8 and ar(K n , S p,q ) for large n, where S p,q is the double star with p + q leaves.

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