Balance in graph-designs

The notion of a balanced incomplete block design (BIBD) was generalized by Hell and Rosa to that of a balanced Γ-design on v vertices. For a given graph Γ, a Γ-design on v vertices is simply a Γ-decomposition of the complete graph Kv. A Γ-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 Γ-designs. Some recent contributions on the following problems are considered: firstly, for each balanced-type condition determining the corresponding spectrum for a given graph Γ; secondly, in case some balanced-type spectra coincide for a given graph Γ, checking if the corresponding classes of balanced-type Γ-designs coincide as well.