Archimedean graph designs

We prove that the complete graph K v can be decomposed into truncated cuboctahedra if v ≡ 1 (mod 144), into icosidodecahedra if v ≡ 1 (mod 120), into rhombicosidodecahedra if v ≡ 1 (mod 240), into truncated icosahedra if v ≡ 1 (mod 180), into truncated dodecahedra if v ≡ 1 (mod 180), into truncated icosidodecahedra if v ≡ 1 (mod 360), into snub cubes if v ≡ 1 (mod 120), and into pseudo-rhombicuboctahedra if and only if v ≡ 1 or 33 (mod 96).