Spanning trees with few branch vertices

TL;DR

This paper proves that large enough connected graphs with high minimum degree contain spanning trees with a limited number of branch vertices, solving a problem motivated by optical network design.

Contribution

It establishes a tight bound on minimum degree ensuring the existence of spanning trees with few branch vertices in connected graphs.

Findings

For all s ≥ 1, graphs with minimum degree ≥ (1/(s+3)+o(1))n contain such spanning trees.

The result is asymptotically optimal.

It addresses a problem motivated by optical network optimization.

Abstract

A branch vertex in a tree is a vertex of degree at least three. We prove that, for all $s\geq 1$, 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, Ku\u{z}el, Li and Ryj\'a\u{c}ek, which was originally motivated by an optimization problem in the design of optical networks.