[[abstract]]For a given integer , a balanced-coloring of a graph  is a mapping  such that  for , where  for . The balanced -decomposition number  of  is the minimum integer  with the property that for any balanced -coloring  there is a partition  such that  induces a connected subgraph with  and  for  and . In this paper, we determine  for some graphs of high connectivity, trees and complete multipartite graphs.[[notice]]補正完