Truncated Tetrahedron, Octahedron and Cube Designs

Abstract

We prove that the complete graph $K_v$ can be decomposed into truncated tetrahedra if and only if $v\equiv 1\text{ or }28(\mathrm{mod}36)$, into truncated octahedra if and only if $v\equiv 1\text{ or }64(\mathrm{mod}72)$, and into truncated cubes if and only if $v\equiv 1\text{ or }64(\mathrm{mod}72)$.