Successive minimum spanning trees

In a complete graph Kn with edge weights drawn independently from a uniform distribution U(0,1) (or alternatively an exponential distribution Exp(1)), let T1 be the MST (the spanning tree of minimum weight) and let Tk be the MST after deletion of the edges of all previous trees Ti, i<k. We show that each tree's weight w(Tk) converges in probability to a constant γk with 2k−2k−−√<γk<2k+2k−−√, and we conjecture that γk=2k−1+o(1). The problem is distinct from that of Frieze and Johansson (2018), finding k MSTs of combined minimum weight, and for k=2 ours has strictly larger cost. Our results also hold (and mostly are derived) in a multigraph model where edge weights for each vertex pair follow a Poisson process; here we additionally have E(w(Tk))→γk. Thinking of an edge of weight w as arriving at time t=nw, Kruskal's algorithm defines forests Fk(t), each initially empty and eventually equal to Tk, with each arriving edge added to the first Fk(t) where it does not create a cycle. Using tools of inhomogeneous random graphs we obtain structural results including that C1(Fk(t))/n, the fraction of vertices in the largest component of Fk(t), converges in probability to a function ρk(t), uniformly for all t, and that a giant component appears in Fk(t) at a time t=σk. We conjecture that the functions ρk tend to time translations of a single function, ρk(2k+x)→ρ∞(x) as k→∞, uniformly in x∈R. Simulations and numerical computations give estimated values of γk for small k, and support the conjectures just stated.