The metamorphosis of λ-fold block designs with block size four into maximum packings of λKn with kites

Abstract

Let (X, B) be a λ-fold block design with block size four and define sets B(K) and E(K₄\ K) as follows: for each block b ε B, remove a path of length two, obtain a kite (a triangle with a tail), and place the kites in B(K) and the paths of length 2 in E(K₄\ K). If we can reassemble the edges belonging to E(K₄\ K) into a collection of kites E(K) with leave L, then (X, B(K) ∪ E(K), L) is a packing of λK _n with kites. If |L| is as small as possible, then (X, B(K) ∪ E(K), L) is called a metamorphosis of the λ-fold block design (X, B) into a maximum packing of λK_n with kites. In this paper we give a complete solution of the metamorphosis problem for λ-fold block designs with block size four into a maximum packing of λK_n with kites for all λ. 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 λK_n with kites.