The metamorphosis of λ-fold block designs with block size four into a maximum packing of λKn with 4-cycles

Let (X,B) be a λ-fold block design with block size four and define sets B(C) and E(K4⧹C) as follows: for each block b∈B, partition b into a 4-cycle and a pair of disjoint edges and place the 4-cycle in B(C) and the 2 disjoint edges in E(K4⧹C). If we can reassemble the edges belonging to E(K4⧹C) into a collection of 4-cycles E(C) with leave L, then (X,B(C)∪E(C),L) is a packing of λKn with 4-cycles and is called a metamorphosis of the λ-fold block design (X,B). In this paper we give a complete solution of the metamorphosis problem for λ-fold block designs into maximum packings of λKn with 4-cycles for all λ (with the possible exception of λ=1, n=37, and leave 2 disjoint triangles). That is, for each λ we determine the set of all n such that there exists a λ-fold block design of order n having a metamorphosis into a maximum packing of λKn with 4-cycles.