Spanning trees with few branch vertices

Abstract

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\ge 1$, every connected graph on $n$ vertices with minimum degree at least $(\frac{1}{s+3}+o(1))n$ contains a spanning tree having at most $s$ branch vertices. Asymptotically, this is best possible and solves, in less general form, a problem of Flandrin, Kaiser, Kuuzel, Li and Ryjáucek, which was originally motivated by an optimization problem in the design of optical networks.

20 pages, 2 figures, to appear in SIAM J. of Discrete Math