Anti-Ramsey numbers in complete split graphs

A subgraph of an edge-coloured graph is rainbow if all of its edges have different colours. For graphs G and H the anti-Ramsey number ar(G,H) is the maximum number of colours in an edge-colouring of G with no rainbow copy of H. The notion was introduced by Erdős, Simonovits and V. Sós and studied in case G = Kn. Afterwards exact values or bounds for anti-Ramsey numbers ar(Kn, H) were established for various H among others by Alon, Jiang & West, Montellano-Ballesteros & Neumann-Lara, Schiermeyer. There are also results concernig bipartite graphs, cubes or product of cycles as G obtained by Axenovich, Li, Montellano-Ballesteros, Schiermeyer and others. In the talk we give the survey of these results and also there will be presented numerous results with a complete split graph Kn + Km as the host graph G.