Balanced and strongly balanced PkPk-designs

Given a graph GG, a G-decomposition   of the complete graph KvKv is a set of graphs, all isomorphic to GG, whose edge sets partition the edge set of KvKv. A GG-decomposition of KvKv is also called a G-design and the graphs of the partition are said to be the blocks  . A GG-design is said to be balanced   if the number of blocks containing any given vertex of KvKv is a constant.In this paper the concept of strongly balanced G-design is introduced and strongly balanced path-designs are studied. Furthermore, we determine the spectrum of those path-designs which are balanced, but not strongly balanced.