论文标题
在边彩色图中的彩虹循环上的注意
Note on rainbow cycles in edge-colored graphs
论文作者
论文摘要
令$ g $是带有边颜色$ c $的订单$ n $的图表，让$δ^c（g）$表示最低颜色度为$ g $。如果$ f $的所有边缘都具有成对的颜色，则$ g $的子图$ f $ of $ g $。边缘色图的彩虹周期有很多结果。在本文中，我们表明（i）如果$δ^c（g）> \ frac {3n-3} {4} $，则$ g $的每个顶点都包含在彩虹三角形中； （ii）$δ^c（g）> \ frac {3n} {4} $，那么$ g $的每个顶点都包含在彩虹$ C_4 $中； （iii）如果$ g $完成，$ n \ geq 8k-18 $和$Δ^c（g）> \ frac {n-1} {2} {2}+k $，则$ g $包含一个长度至少$ k $的彩虹周期。还发现并纠正了先前出版物中的一些差距。
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $δ^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $δ^c(G)>\frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $δ^c(G)>\frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $n\geq 8k-18$ and $δ^c(G)>\frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of length at least $k$. Some gaps in previous publications are also found and corrected.
