论文标题
边缘彩色完整图的正确顶点 - 无关节单色三角形
Proper vertex-pancyclicity of edge-colored complete graphs without joint monochromatic triangles
论文作者
论文摘要
在边缘色图$（g，c）$中，令$ d^c（v）$表示边缘事件上的颜色数量，带有$ g $ $ g $的顶点$ v $和$δ^c（g）$表示所有Vertices $ v \ in V（g）$中的最低值为$ d^c（v）$。如果周期的任何两个相邻边缘具有不同的颜色，则$（g，c）$的周期称为正确。 $ n \ geq 3 $顶点上的边缘色图$（g，c）$如果每个$（g，c）$的每个顶点均包含在每个$ \ ell $的适当循环中，则称为$（g，c）$的每个顶点，带有$ 3 \ ell $的适当循环。 Fujita和Magnant猜想，$ n \ geq 3 $ dertices上的每个边彩色完整图，带有$δ^c（g）\ geq \ frac {n+1} {2} {2} $是正确的vertex-pancyclic。 Chen，Huang和Yuan通过添加$（G，C）$不包含任何单色三角形的额外条件来部分解决这种猜想。在本文中，我们表明，如果边缘色完整图没有关节单色三角形，则此猜想是正确的。
In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $δ^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with $δ^c(G)\geq \frac{n+1}{2}$ is properly vertex-pancyclic. Chen, Huang and Yuan partially solve this conjecture by adding an extra condition that $(G,c)$ does not contain any monochromatic triangle. In this paper, we show that this conjecture is true if the edge-colored complete graph contain no joint monochromatic triangles.
