Computing the anti-Ramsey number for trees in complete tripartite graph

•Given an edge-colored graph G, if any two edges of G receive distinct colors, then we call G a rainbow graph. The anti-Ramsey number AR(G;F) is the maximum integer k in a k-edge-coloring of G that there is no any rainbow graph in F.•An interesting topic about edge-colored graphs is to study the order of monochromatic or rainbow component. Here we focus on anti-Ramsey problem on rainbow component. This equals to study the existence of rainbow trees. We call a tree of order k+1 a k-tree, since it just contains k edges.•The anti-Ramsey problem on k-trees obviously received less attention. The value of anti-Ramsey number of k-trees in Kn has been determined by Jiang and West. Later, Jin and Li considered its anti-Ramsey number in complete bipartite graph. Recently, Zhang and Dong gave a formula to determine the value of AR(G; k-trees), where G is a complete multipartite graph of order n, for 4n−25≤k≤n−1. When k is small, it is difficult to determine the exact value. In this paper, we aim to determine the exact value of AR(G; k-trees) when G is a balanced complete tripartite graph, especially for k much less than the order of host graph. The anti-Ramsey number AR(K,F) is the largest color number in an edge-coloring of K that contains no rainbow copies of graphs in the family F. In this paper, we determine the exact value of the anti-Ramsey number for trees in a balanced complete tripartite graph.