An edge-colored graph is called rainbow if all its edges are colored distinct. The anti-Ramsey number of a graph family \({\mathcal {F}}\) in the graph G, denoted by \(AR{(G,{\mathcal {F}})}\), is the maximum number of colors in an edge-coloring of G without rainbow subgraph in \({\mathcal {F}}\). The anti-Ramsey number for the short cycle \(C_3\) has been determined in a few graphs. Its anti-Ramsey number in the complete graph can be easily obtained from the lexical edge-coloring. Gorgol considered the problem in complete split graphs which contains complete graphs as a subclass. In this paper, we study the problem in the complete multipartite graph which further enlarges the family of complete split graphs. The anti-Ramsey numbers for \(C_3\) and \(C_3^{+}\) in complete multipartite graphs are determined. These results contain the known results for \(C_3\) and \(C_3^{+}\) in complete and complete split graphs as corollaries.

如果所有边缘的颜色都不同，则边缘着色的图称为彩虹。图G中图族\（{\ mathcal {F}} \）的反Ramsey数由\（AR {（G，{\ mathcal {F}}）} \）表示为最大值。\（{\ mathcal {F}} \）中没有彩虹子图的G的边缘着色中的颜色集合。短周期\（C_3 \）的反Ramsey数已在一些图表中确定。完整图形中的反Ramsey数可以从词汇边缘着色中轻松获得。Gorgol在完全拆分图中考虑了该问题，该拆分图中包含完整图作为子类。在本文中，我们研究了完全多部分图中的问题，这进一步扩大了完全分裂图的族。确定完整多部分图中\（C_3 \）和\（C_3 ^ {+} \）的反Ramsey数。这些结果包含\（C_3 \）和\（C_3 ^ {+} \）的已知结果，作为完整推论和完整分裂图。