A hierarchy of balanced graph-designs

Decompositions of the complete graph K_v into subgraphs, all of which are isomorphic to some given non-regular graph G are considered. The decompositions are required to have the additional property that each vertex occurs a constant number of times as a vertex of given degree in the subgraphs of the decomposition. These decompositions are said to be degree-balanced G-designs. General properties of degree-balanced G-designs are studied and the spectrum of degree-balanced G-designs is determined when G is a bowtie. Moreover, for each v in this spectrum, there exists a bowtie design on v vertices which is not degree-balanced.