论文标题
边缘色完整图中的彩虹三角形
Rainbow triangles in edge-colored complete graphs
论文作者
论文摘要
让$ g $是带有边颜色$ c $的订单$ n $的图表，让$δ^c（g）$表示$ g $的最低颜色度。如果任何两个边缘的$ f $具有不同的颜色，则$ g $的子图$ f $ of $ g $。现有关于边缘色完整图中彩虹三角形的文献中有很多结果。 Fujita和Magnant表明，对于边彩的完整图$ g $ of订单$ n $，如果$δ^c（g）\ geq \ frac \ frac {n+1} {2} $，则$ g $的每个顶点都包含在Rainbow Triangle中。在本文中，我们表明，如果$δ^c（g）\ geq \ frac {n+k} {2} $，则$ g $的每个顶点都包含在至少$ k $ rainbow Triangles中，这可以看作是其结果的概括。 Li表明，对于边缘颜色的$ g $ of Order $ n $，如果$δ^c（g）\ geq \ frac \ frac {n+1} {2} {2} $，则$ g $包含一个彩虹三角形。我们表明，如果$ g $已完成，并且$δ^c（g）\ geq \ frac {n} {2} $，则$ g $包含一个彩虹三角，并且界限很清晰。 Hu等。表明，对于边缘颜色的图形$ g $ of订单$ n \ geq 20 $，如果$δ^c（g）\ geq \ frac {n+2} {2} {2} $，则$ g $包含两个vertex-disjoint rainbow triangles。我们表明，如果$ g $与订单$ n \ geq 8 $和$δ^c（g）\ geq \ frac {n+1} {2} $一起完成，则$ g $包含两个vertex-Disjoint Rainbow Triangles。此外，我们改善了Hu等人的结果。从$ n \ geq 20 $到$ n \ geq 7 $，是最好的。
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 any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $δ^c(G)\geq \frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $δ^c(G)\geq \frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k$ rainbow triangles, which can be seen as a generalization of their result. Li showed that for an edge-colored graph $G$ of order $n$, if $δ^c(G)\geq \frac{n+1}{2}$, then $G$ contains a rainbow triangle. We show that if $G$ is complete and $δ^c(G)\geq \frac{n}{2}$, then $G$ contains a rainbow triangle and the bound is sharp. Hu et al. showed that for an edge-colored graph $G$ of order $n\geq 20$, if $δ^c(G)\geq \frac{n+2}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. We show that if $G$ is complete with order $n\geq 8$ and $δ^c(G)\geq \frac{n+1}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. Moreover, we improve the result of Hu et al. from $n\geq 20$ to $n\geq 7$, the best possible.
