Balance in graph-designs

The notion of a balanced incomplete block design (BIBD) was generalized by Hell and Rosa to that of a balanced G-design on v vertices. For a given graph G, a G-design on v vertices is simply a G-decomposition of the complete graph K_v. A G-design is said to be balanced if there exists a constant r such that for each point x the number of blocks containing x is equal to r. Different conditions of ``balanced'' type can be defined by assuming that some other local parameter is a constant. In this vein one can define, for instance, orbit-balanced resp. degree-balanced G-designs. Some recent contributions on the following problems are considered: firstly, for each balanced-type condition determining the corresponding spectrum for a given graph G; secondly, in case some balanced-type spectra coincide for a given graph G, checking if the corresponding classes of balanced-type G-designs coincide as well.