Abstract A spanning circuit in a graph is defined as a closed trail visiting each vertex of the graph. A compatible spanning circuit in an edge-colored graph refers to a spanning circuit in which each pair of edges traversed consecutively along the spanning circuit has distinct colors. As two extreme cases, sufficient conditions for the existence of compatible Hamilton cycles and compatible Euler tours have been obtained in previous literature. In this paper, we first establish sufficient conditions for the existence of compatible spanning circuits visiting each vertex exactly k times, for every feasible integer k , in edge-colored complete graphs and complete equipartition r -partite graphs. We also provide sufficient conditions for the existence of compatible spanning circuits visiting each vertex v at least ⌊ ( d ( v ) − 1 ) ∕ 2 ⌋ times in edge-colored graphs satisfying Ore-type degree conditions.

中文翻译：

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边彩色图中的兼容生成电路

摘要 图中的生成回路被定义为访问图的每个顶点的闭合路径。边着色图中的兼容生成电路是指沿着生成电路连续遍历的每对边具有不同颜色的生成电路。作为两个极端情况，之前的文献已经获得了兼容哈密顿圈和兼容欧拉环存在的充分条件。在本文中，我们首先为边着色完全图和完全均分 r 部分图中的每个可行整数 k ，为每个顶点恰好 k 次访问的兼容生成电路的存在建立充分条件。