The Existence and Construction of ( K 5 ∖ e )‐Designs of Orders 27, 135, 162, and 216

The problem of the existence of a decomposition of the complete graph K n into disjoint copies ofK 5 ∖ 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 ( K 27 , K 5 ∖ e ) ‐design. I show that | A u t ( Γ ) | divides 2 k 3 for some k ≥ 0 and that S y m ( 3 ) ≰ A u t ( Γ ) . I construct ( K 27 , K 5 ∖ e ) ‐designs by prescribing Z 6 as an automorphism group, and show that up to isomorphism there are exactly 24 ( K 27 , K 5 ∖ e ) ‐designs with Z 6 as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed Z 6 . Finally, the existence of ( K 5 ∖ 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.