Anti-Ramsey numbers of graphs with some decomposition family sequences

TL;DR

This paper introduces the concept of decomposition family sequences of graphs to determine anti-Ramsey numbers for various graph families, including the Petersen graph and unions of cliques, and characterizes extremal colorings.

Contribution

It extends the decomposition family concept to sequences, enabling the determination of anti-Ramsey numbers for new graph classes and characterizing extremal colorings.

Findings

Determined anti-Ramsey numbers for the Petersen graph.

Extended the decomposition family to sequences.

Characterized extremal colorings for new graph families.

Abstract

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.