Star-factors in graphs of high degree

TL;DR

This paper proves that graphs with high minimum degree contain large star-structured spanning forests, improving previous bounds and approaching optimality up to lower order terms.

Contribution

It establishes a new bound on the size of star components in spanning forests of high-degree graphs, refining earlier results by Alon and Wormald.

Findings

Graphs with large minimum degree contain spanning forests with large star components.

The size of star components is at least rom the abstract, likely or some function of d.

The result is optimal up to lower order terms.

Abstract

We prove that every graph with sufficiently large minimum degree $d$ contains a spanning forest in which every component is a star of size at least $\sqrt{d} - \tilde O(d^{1/4})$. This improves the result of Alon and Wormald and is optimal up to the lower order term.