Anti-Ramsey numbers of graphs with some decomposition family sequences

       For a given graph H, the anti-Ramsey number of H is the maximum number of colors in an edge-coloring of a complete graph which does not contain a rainbow copy of H. In this paper, we extend the decomposition family of graphs to the decomposition family sequence of graphs and show that K ₅ is determined by its decomposition family sequence. Based on this new graph notation, we determine the anti-Ramsey numbers for new families of graphs, including the Petersen graph, vertex-disjoint union of cliques, etc., and characterize the extremal colorings.