In a complete graph with independent uniform (or exponential) edge weights, let be the minimum-weight spanning tree (MST), and the MST after deleting the edges of all previous trees. We show that each tree's weight converges in probability to a constant , with , and we conjecture that . The problem is distinct from Frieze and Johansson's minimum combined weight of k edge-disjoint spanning trees; indeed, . With an edge of weight w “arriving” at time , Kruskal's algorithm defines forests , initially empty and eventually equal to , each edge added to the first possible . Using tools of inhomogeneous random graphs we obtain structural results including that the fraction of vertices in the largest component of converges to some . We conjecture that the functions tend to time translations of a single function.

中文翻译：

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连续最小生成树

在具有独立均匀（或指数）边权重的完整图中，令为最小权重生成树（MST），以及删除所有先前树的边后的 MST。我们证明了每棵树的权重在概率上收敛到一个常数，我们推测。该问题与 Frieze 和 Johansson 的 k 边不相交生成树的最小组合权重不同；确实，. 随着权重w的边在时间“到达” ，Kruskal 的算法定义了森林，最初是空的，最终等于，每条边都添加到第一个可能的边上. 使用非齐次随机图的工具，我们获得了结构结果，包括最大分量中的顶点分数收敛到一些。我们推测这些函数倾向于对单个函数的时间翻译。