High degree graphs contain large-star factors

TL;DR

This paper proves that graphs with high minimum degree contain large star forests, with each component being significantly large, addressing a previously open problem in graph theory.

Contribution

It establishes a lower bound on the size of star components in spanning forests of high-degree graphs, solving a problem posed by J. Kratochvil.

Findings

Graphs with minimum degree d contain large star forests.

Each star component has size at least proportional to (d/log d)^{1/3}.

Addresses an open problem in graph theory.

Abstract

We show that any finite simple graph with minimum degree $d$ contains a spanning star forest in which every connected component is of size at least $\Omega((d/\log d)^{1/3})$. This settles a problem of J. Kratochvil.