Abstract Let G be a (not necessarily properly) edge-colored graph. A compatible spanning circuit in G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. As two extremal cases, the existence of compatible (i.e., properly edge-colored) Hamilton cycles and compatible Euler tours have been studied extensively. More recently, sufficient conditions for the existence of compatible spanning circuits visiting each vertex v of G at least ⌊ ( d ( v ) − 1 ) ∕ 2 ⌋ times in graphs satisfying Ore-type degree conditions have been established. In this paper, we continue the research on sufficient conditions for the existence of compatible spanning circuits visiting each vertex at least a specified number of times. We respectively consider graphs satisfying Fan-type degree conditions, graphs with a high edge-connectivity, and the asymptotical existence of such compatible spanning circuits in random graphs.

中文翻译：

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边色图中几乎欧拉兼容的生成电路

摘要 令 G 是（不一定正确）边着色图。G 中的兼容生成电路是包含 G 的所有顶点的封闭路径，其中任意两条连续遍历的边具有不同的颜色。作为两个极值情况，已经广泛研究了兼容（即正确边缘着色）的汉密尔顿循环和兼容的欧拉旅行的存在。最近，已经建立了在满足 Ore-type 度条件的图中，存在访问 G 的每个顶点 v 至少 ⌊ ( d ( v ) − 1 ) ∕ 2 ⌋ 次的兼容生成电路的充分条件。在本文中，我们继续研究存在至少访问每个顶点指定次数的兼容生成电路的充分条件。我们分别考虑满足 Fan-type 度条件的图，