Graph Divisible Designs and Packing Constructions

We introduce a generalization of group divisible designs and offer example applications to challenging problems in design theory. The generalization considers edge-decompositions of joins of arbitrary graphs, whereas group divisible designs handle only joins of edgeless graphs. Our example constructions include: (1) optimal packings with block size five for the previously unsettled congruence class $$v \equiv 13 \pmod {20}$$v≡13(mod20); (2) an optimal grooming with with ratio seven for the previously unsettled congruence class $$v \equiv 56 \pmod {84}$$v≡56(mod84); and (3) a constructive ‘quadratic’ embedding of partial designs with block size four.