Completeness of vertex- and edge-transitive 3D nets with nearest-neighbor edges

Establish whether the set of 20 known vertex- and edge-transitive three-dimensional nets obtained by connecting nearest-neighbor vertices in their highest-symmetry embeddings is complete; specifically, prove that no additional such nets exist or construct explicit counterexamples to demonstrate incompleteness. This classification question underpins systematic searches for frustrated magnetic states on high-symmetry nets.

Background

The paper focuses on vertex- and edge-transitive nets because they uniquely allow frustration without fine-tuning, making them central for discovering unconventional magnetic states. The authors consider 20 such nets formed by nearest-neighbor connections (plus one next-nearest neighbor case, lcx), guided by RCSR nomenclature and high-symmetry embeddings.

Despite extensive searches reported in the literature, a formal proof that these 20 nets exhaust all possible vertex- and edge-transitive 3D nets with nearest-neighbor edges is lacking. Resolving this would solidify the foundational topology set used for magnetism studies and materials design.