Metamorphosis of simple twofold triple systems into maximum twofold (K4−e)-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 B1,B2∈B satisfying |B1∩B2|=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 (K4−e)[x,y,z−w]. Let C be the collection of (K4−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⌋(K4−e)s, then (X,C∪D) is a maximum twofold (K4−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 (K4−e)-packing of order v if and only if v≡0,1(mod3) and v≥4 with two exceptions of v=6,7 and one possible exception of v=18.