Star arboricity

A star forest 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 ~).