Anti-Ramsey Number of Triangles in Complete Multipartite Graphs

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.