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For a given graph $H$, the anti-Ramsey number of $H$ is the maximum number ofcolors in an edge-coloring of a complete graph which does not contain a rainbowcopy of $H$. In this paper, we extend the decomposition family of graphs to thedecomposition family sequence of graphs and show that $K_5$ is determined byits decomposition family sequence. Based on this new graph notation, wedetermine the anti-Ramsey numbers for new families of graphs, including thePetersen graph, vertex-disjoint union of cliques, etc., and characterize theextremal colorings.

Anti-Ramsey numbers of graphs with some decomposition family sequences