Summary: High degree graphs contain large-star factors
Dedicated to L´aszl´o Lov´asz, for his 60th birthday
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 ((d/ log d)1/3). This
settles a problem of Havet, Klazar, Kratochvil, Kratsch and Liedloff.
This paper is dedicated to Laci Lov´asz, for his 60th birthday. It settles a prob-
lem presented by Jan Kratochvil at the open problems session of the meeting Building
Bridges, which took place in Budapest in August 2008, celebrating this birthday. The
Lov´asz Local Lemma is applied extensively throughout the proof. This work is there-
fore a typical example illustrating the immense influence of Laci, who not only provided
the community with powerful tools and techniques, but also stimulated research by his
books, lectures and organization of conferences.
All graphs considered here are finite and simple. A star is a tree with one vertex, the center,
adjacent to all the others, which are leaves. A star factor of a graph G is a spanning forest of
G in which every connected component is a star. It is easy to see that any graph with positive