The Existence and Construction of (K5e)-Designs of Orders 27, 135, 162, and 216



The problem of the existence of a decomposition of the complete graph inline image into disjoint copies of inline image has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a inline image-design. I show that inline image divides 2k3 for some inline image and that inline image. I construct inline image-designs by prescribing inline image as an automorphism group, and show that up to isomorphism there are exactly 24 inline image-designs with inline image as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed inline image. Finally, the existence of inline image-designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864.