The Existence and Construction of (K5∖e)-Designs of Orders 27, 135, 162, and 216 - Kolotolu, Emre (Journal of combinatorial designs, 2013)
The problem of the existence of a decomposition of the complete graph Kn into disjoint copies of K5∖e 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 (K27,K5∖e)‐design. I show that |Aut(Γ)| divides 2k3 for some k≥0 and that Sym(3)≰Aut(Γ). I construct (K27,K5∖e)‐designs by prescribing Z6 as an automorphism group, and show that up to isomorphism there are exactly 24 (K27,K5∖e)‐designs with Z6 as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed Z6. Finally, the existence of (K5∖e)‐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.