In a complete graph with independent uniform (or exponential) edge weights, let be the minimum-weight spanning tree (MST), and MST after deleting edges of all previous trees. We show that each tree's weight converges in probability to constant , we conjecture . The problem is distinct from Frieze Johansson's minimum combined k edge-disjoint trees; indeed, With an w “arriving” at time Kruskal's ...

We show that for every graph G contains two edge-disjoint spanning trees, we can choose trees T1,T2 of such |dT1(v)−dT2(v)|≤5 all v∈V(G). also prove the more general statement positive integer k, there is a constant ck∈O(logk) k T1,…,Tk satisfying |dTi(v)−dTj(v)|≤ck v∈V(G) and i,j∈{1,…,k}. This resolves conjecture Kriesell.