SPANNING TREES WITH FEW BRANCH VERTICES

摘要

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all s >= 1, every connected graph on n vertices with minimum degree at least (1/s+3 +o(1))n contains a spanning tree having at most s branch vertices. Asymptotically, this is best possible and solves a problem of Flandrin, Kaiser, Kuzel, Li, and Ryjcek, which was originally motivated by an optimization problem in the design of optical networks.