Cuboctahedron designs (Q2708249)

The authors solve the problem of simultaneously decomposing the complete graph \(K_v\) into subgraphs \(K_3\) and \(C_4\). This leads to the construction of a cuboctahedron design and they prove that such a design exists if and only if \(v\equiv 1\) or \(33\pmod{48}\). The proof is by construction: first designs of small orders are produced and then suitable group-divisible designs are used to obtain the infinite families. The exceptional cases are dealt with by ad hoc constructions. Also, the authors briefly review the main known results on the decompositions of the complete graph into ``nice'' subgraphs.