The metamorphosis of λ-fold K4-e designs into maximum packings of λKn with 4-cycles λ≥2

Let K-4 - e = square. If we remove the "diagonal" edge the result is a 4-cycle. Let (X, B) be a lambda-fold K-4 - e design of order n: i.e., a decomposition of lambda K-n into copies of K-4 - e. Let D(B) be the collection of "diagonals" removed from the graphs in B and C-1(B) the resulting collection of 4-cycles. If C-2(B) is a reassembly of these edges into 4-cycles and L is the collection of edges in D(B) not used in a 4-cycle of C-2(B), then (X, C-1(B)boolean OR C-2(B), L) is a packing of lambda K-n with 4-cycles and is called a metamorphosis of (X, B). In Lindner and Tripodi [2005. The metamorphosis of K-4 - e designs into maximum packings of K-n with 4-cycles. Ars Combin. 75, 333-349.] a complete solution is given for the existence problem of K-4 - e designs (lambda = I) having a metamorphosis into a maximum packing of K-n with all possible leaves, The purpose of this paper is the complete solution of the above problem for all values of lambda > 1. (C) 2008 Elsevier B.V. All rights reserved.