Archimedean graph designs

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