A note on random minimum length spanning trees

Abstract

Consider a connected r-regular n-vertex graph G with random independent edge lengths, each uniformly distributed on [0, 1]. Let mst(G) be the expected length of a minimum spanning tree. We show in this paper that if G is sufficiently highly edge connected then the expected length of a minimum spanning tree is ∼ n/rζ(3). If we omit the edge connectivity condition, then it is at most ∼ n/r(ζ(3) + 1).