COMBINATORICA Akad~miai Kiadd -Springer-Verlag Summary: COMBINATORICA Akad~miai Kiadd - Springer-Verlag COMBINATORICA12 (4) (1992) 375-380 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 Collections: Mathematics