论文标题
边缘彩色完整图中长度3和4的循环，并在颜色过渡中受到限制
Cycles of length 3 and 4 in edge-colored complete graphs with restrictions in the color transitions
论文作者
论文摘要
令$ g $是边缘颜色的图，$ g $的步行据说是正确的彩色步行，如果连续的边缘有不同的颜色，包括第一个和最后一个边缘，以防万一步行关闭。令$ h $成为循环的图表。我们会说图形$ g $是$ h $颜色的图，如果存在函数$ c：e（g）\ longrightarrow v（h）$。 $ g $中的路径$（v_1，\ cdots，v_k）$是$ h $ - $ h $ path时（c（v_1v_2），\ cdots，$ $ $ c（v_ {k-1} v_k）$是$ h $的步行路程v_2），\ cdots，c（v_ {k-1} v_k），$ $ c（v_kv_1），c（v_1 v_2））$是$ h $的步行路。因此，$ h $决定在步行中允许哪些颜色过渡，以便成为$ h $ walk。每当$ h $是一个没有循环的完整图形时，$ h $ walk是一个适当的彩色步行，因此$ h $ - walk是一个更一般的概念。在本文中，我们使用$ h $颜色的完整图，并带有辅助图给出的限制。主要定理给出的条件表明，$ h $颜色的完整图中的每个顶点都包含在长度为3的$ H $周期中，并且在$ h $ cycle的长度为4中。由于主要结果，我们获得了一些众所周知的定理。
Let $G$ be an edge-colored graph, a walk in $G$ is said to be a properly colored walk iff each pair of consecutive edges have different colors, including the first and the last edges in case that the walk be closed. Let $H$ be a graph possible with loops. We will say that a graph $G$ is an $H$-colored graph iff there exists a function $c:E(G)\longrightarrow V(H)$. A path $(v_1,\cdots,v_k)$ in $G$ is an $H$-path whenever $(c(v_1v_2),\cdots,$ $c(v_{k-1}v_k))$ is a walk in $H$, in particular, a cycle $(v_1,\cdots,v_k,v_1)$ is an $H$-cycle iff $(c(v_1 v_2),\cdots,c(v_{k-1}v_k),$ $c(v_kv_1), c(v_1 v_2))$ is a walk in $H$. Hence, $H$ decide which color transitions are allowed in a walk, in order to be an $H$-walk. Whenever $H$ is a complete graph without loops, an $H$-walk is a properly colored walk, so $H$-walk is a more general concept. In this paper, we work with $H$-colored complete graphs, with restrictions given by an auxiliary graph. The main theorems give conditions implying that every vertex in an $H$-colored complete graph, is contained in an $H$-cycle of length 3 and in an $H$-cycle of length 4. As a consequence of the main results, we obtain some well-known theorems in the theory of properly colored walks.
