The anti-Ramsey number of $C_3$ and $C_4$ in the complete $r$-partite graphs

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 $ar(K_{n_1,n_2,\dots ,n_r},\{C_3,C_4\}),ar(K_{n_1,n_2,\dots ,n_r},C_3)$ and $ar(K_{n_1,n_2,\dots ,n_r},C_4)$.

12 pages

References in corpus (1)

• Anti-Ramsey numbers of graphs with some decomposition family sequences

Cited by in corpus (1)

• Turán numbers and anti-Ramsey numbers for short cycles in complete $3$-partite graphs