The anti-Ramsey number of $C_{3}$ and $C_{4}$ in the complete $r$-partite graphs

TL;DR

This paper determines the maximum number of colors in edge-colorings of complete r-partite graphs that avoid rainbow triangles and quadrilaterals, extending anti-Ramsey theory to these specific cycles.

Contribution

It provides exact values for the anti-Ramsey numbers of C3 and C4 in complete r-partite graphs, a problem previously not fully addressed.

Findings

Exact formulas for ar(K_{n_1,...,n_r}, {C_3, C_4})

Exact formulas for ar(K_{n_1,...,n_r}, C_3)

Exact formulas for ar(K_{n_1,...,n_r}, C_4)

Abstract

A subgraph of an edge-colored graph is rainbow, if all of its edges have different colors. For a graph $G$ and a family $\mathcal{H}$ of graphs, the anti-Ramsey number $ar(G, \mathcal{H})$ is the maximum number $k$ such that there exists an edge-coloring of $G$ with exactly $k$ colors without rainbow copy of any graph in $\mathcal{H}$. In this paper, we study the anti-Ramsey number of $C_{3}$ and $C_{4}$ in the complete $r$-partite graphs. For $r\ge 3$ and $n_{1}\ge n_{2}\ge \cdots\ge n_{r}\ge 1$, we determine $ ar(K_{n_{1}, n_{2}, \ldots, n_{r}},\{C_{3}, C_{4}\}), ar(K_{n_{1}, n_{2}, \ldots, n_{r}}, C_{3})$ and $ar(K_{n_{1}, n_{2}, \ldots, n_{r}}, C_{4})$.