The metamorphosis of block designs with block size four into (K4 / e)-designs

Denote K4 / e (= the complete undirected graph K4 with one edge removed) by diagonal inside box sign. A diagonal inside box sign -design of order n is a pair (X, K), where K is a collection of edge disjoint copies of diagonal inside box sign which partitions the edge set of Kn with vertex set X. Let (X, B) be a block design with block size 4. If we remove one edge from each block in B, we obtain a partial diagonal inside box sign -design. If the deleted edges can be arranged into copies of diagonal inside box sign the result is a diagonal inside box sign -design, called a metamorphosis of the block design (X, B). In this paper we prove that the spectrum for block designs with block size 4 having a metamorphosis into a diagonal inside box sign -design is precisely the set of all n ≡ 1, 16, 25, or 40 (mod 60).