A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.

边色图G（不一定正确）中的兼容生成电路是包含G的所有顶点的闭合轨迹其中任意两个连续遍历的边缘具有不同的颜色。在先前的文献中已经考虑了存在极值相容跨越电路（即，相容汉密尔顿循环和欧拉环）的充分条件，以及用于寻找相容欧拉环的多项式时间算法。最近，已经建立了在特定的边缘彩色图中存在更通用的兼容生成电路的充分条件。在本文中，我们从算法的角度考虑了（更通用的）兼容生成电路的存在。我们首先显示确定边缘着色的连通图是否包含兼容的生成电路是一个NP完全问题。接下来，我们描述两种用于在边色完整图中找到兼容的生成电路的多项式时间算法。