The metamorphosis of λλ-fold K4-eK4-e designs into maximum packings of λKnλKn with 4-cycles, λ⩾2λ⩾2

Let K4-e=. If we remove the “diagonal” edge the result is a 4-cycle. Let (X,B)(X,B) be a λλ-fold K4-eK4-e design of order n  ; i.e., a decomposition of λKnλKn into copies of K4-eK4-e. Let D(B)D(B) be the collection of “diagonals” removed from the graphs in B   and C1(B)C1(B) the resulting collection of 4-cycles. If C2(B)C2(B) is a reassembly of these edges into 4-cycles and L   is the collection of edges in D(B)D(B) not used in a 4-cycle of C2(B)C2(B), then (X,C1(B)∪C2(B),L)(X,C1(B)∪C2(B),L) is a packing of λKnλKn with 4-cycles and is called a metamorphosis   of (X,B)(X,B). In Lindner and Tripodi [2005. The metamorphosis of K4-eK4-e designs into maximum packings of KnKn with 4-cycles. Ars Combin. 75, 333–349.] a complete solution is given for the existence problem of K4-eK4-e designs (λ=1λ=1) having a metamorphosis into a maximum packing   of KnKn with all possible leaves. The purpose of this paper is the complete solution of the above problem for all   values of λ>1λ>1.