论文标题
$ c_ {3} $和$ c_ {4} $的反斑点号
The anti-Ramsey number of $C_{3}$ and $C_{4}$ in the complete $r$-partite graphs
论文作者
论文摘要
如果其所有边缘都有不同的颜色，则彩虹的边彩色图是彩虹。对于图形$ g $和一个家庭$ \ MATHCAL {H} $的图形，抗Ramsey Number $ ar（g，\ nathcal {h}）$是最大数字$ k $，因此存在$ g $的边缘色，$ k $ agry $ k $ colors coles coles colors colors colors colors colors $ \ mathcal中的任何图形副本，nmathcal calcal calcal {h h} $。在本文中，我们研究了完整的$ r $ - 分段图中的$ c_ {3} $的反拉姆西数和$ c_ {4} $。对于$ r \ ge 3 $和$ n_ {1} \ ge n_ {2} \ ge \ cdots \ ge n_ {r} \ ge 1 $，我们确定$ ar（k_ {n_ {1}，n_ {1}，n_ {2}，n_ {2}，\ ldots，\ ldots，n_ {r}} ar（k_ {n_ {1}，n_ {2}，\ ldots，n_ {r}}，c_ {3}）$和$ ar（k_ {n_ {n_ {1}，n_ {2}，n_ {2}，\ ldots，\ ldots，n_ {r}}，c_ {r}}，c_ c_ {4}）$。
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 $ar(K_{n_{1}, n_{2}, \ldots, n_{r}}, C_{4})$.
