On the minimum number of bond-edge types and tile types: An approach by edge-colorings of graphs

The purpose of this paper is twofold. On one hand, we want to describe from a new graph theory perspective the self-assembly of DNA structures with branched junction molecules having flexible arms. On the other hand, we employ edge-colorings and graph decompositions to study the well-known problem of determining the minimum number of bond-edge types and tile types, which are graph invariants appearing in this context. We provide a strategy that can be applied to arbitrary graphs for obtaining upper bounds for these graph invariants.