Successive minimum spanning trees

Abstract

In a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$ (or alternatively an exponential distribution $\operatorname{Exp}(1)$), let $T_1$ be the MST (the spanning tree of minimum weight) and let $T_k$ be the MST after deletion of the edges of all previous trees $T_i$, $i<k$. We show that each tree's weight $w(T_k)$ converges in probability to a constant $γ_k$ with $2k-2\sqrt k <γ_k<2k+2\sqrt k$, and we conjecture that $γ_k = 2k-1+o(1)$. 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 $\mathbb E(w(T_k)) \to γ_k$. Thinking of an edge of weight $w$ as arriving at time $t=n w$, Kruskal's algorithm defines forests $F_k(t)$, each initially empty and eventually equal to $T_k$, with each arriving edge added to the first $F_k(t)$ where it does not create a cycle. Using tools of inhomogeneous random graphs we obtain structural results including that $C_1(F_k(t))/n$, the fraction of vertices in the largest component of $F_k(t)$, converges in probability to a function $ρ_k(t)$, uniformly for all $t$, and that a giant component appears in $F_k(t)$ at a time $t=σ_k$. We conjecture that the functions $ρ_k$ tend to time translations of a single function, $ρ_k(2k+x)\toρ_\infty(x)$ as $k \to \infty$, uniformly in $x\in \mathbb R$. Simulations and numerical computations give estimated values of $γ_k$ for small $k$, and support the conjectures just stated.