 
Summary: COMBINATORICA
Akad~miai Kiadd  SpringerVerlag
COMBINATORICA12 (4) (1992) 375380
STAR ARBORICITY
NOGA ALON, COLIN McDIARMID and BRUCE REED
ReceivedSeptember 11, 1989
A starforest is a forest all of whose components are stars. The star arboricity,st(G) of a graph
G is the minimum number of star forests whose union covers all the edges of G. The arboricity,
A(G), of a graph G is the minimum number of forests whose union covers all the edges of G. Clearly
st(G) >A(G). In fact, Algor and Alon have given examples which show that in some cases st(G)
can be as large as A(G)+ f~(log/k) (where s is the maximum degree of a vertex in G). We show
that for any graph G, st(G) <_A(G) +O(log~).
1. Introduction
All graphs considered here are finite and simple. For a graph H, let E(H) denote
the set of its edges, and let V(H) denote the set of its vertices.
A star is a tree with at most one vertex whose degree is not one. A star forest
is a forest whose connected components are stars. The star arboricity of a graph G,
denoted st(G), is the minimum number of star forests whose union covers all edges
of G. The arboricity of G, denoted A(G) is the minimum number of forests needed
to cover all edges of G. Clearly, st(G) > A(G) by definition. Furthermore, it is easy
