Metamorphosis of simple twofold triple systems into maximum twofold -packings

Let ( X , B ) be a simple twofold triple system of order v . For every x , y ∈ X , x ≠ y , the pair { x , y } is contained in exactly two different triples, say, { x , y , z } and { x , y , w } . Any two blocks B 1 , B 2 ∈ B satisfying | B 1 ∩ B 2 | = 2 form a matched pair. Suppose that there is a partition of B into | B | / 2 matched pairs. If we replace the double edge { x , y } with its corresponding single edge { x , y } from a matched pair { x , y , z } , { x , y , w } , we have a ( K 4 − e ) [ x , y , z − w ] . Let C be the collection of ( K 4 − e ) s obtained by replacing the double edge of each matched pair of B with its corresponding single edge, and F be the collection of the deleted edges. If F can be reassembled into a collection D of ⌊ v ( v − 1 ) / 30 ⌋ ( K 4 − e ) s, then ( X , C ∪ D ) is a maximum twofold ( K 4 − e ) -packing of order v . We call ( X , C ∪ D ) a metamorphosis of the simple twofold triple system ( X , B ) . In this paper, we show that there exists a metamorphosis of a simple twofold triple system of order v into a maximum twofold ( K 4 − e ) -packing of order v if and only if v ≡ 0 , 1 ( mod 3 ) and v ≥ 4 with two exceptions of v = 6 , 7 and one possible exception of v = 18 .