On spanning trees with high internal degree

TL;DR

This paper investigates the existence of spanning trees with high internal degree in graphs with large minimum degree, providing a counterexample to a previously posed conjecture about such trees.

Contribution

The paper presents a simple counterexample demonstrating that not all connected graphs with high minimum degree have spanning trees with high internal degree.

Findings

Counterexample shows the conjecture is false

High minimum degree does not guarantee high internal degree in spanning trees

Clarifies limitations in spanning tree structures in dense graphs

Abstract

Alon and Wormald showed that any graph with minimum degree d contains a spanning star forest in which every connected component is of size at least \Omega((d/\log d)^{1/3}). They asked if any connected graph with minimum degree at least d has a spanning tree in which every internal vertex has degree at least cd/\log d, for some absolute constant c > 0. We give a simple example showing that this is not the case.