The metamorphosis of λ-fold block designs with block size four into λ-fold (K4 \ e)-systems, λ ≥ 2
Abstract
Denote K4 \ e (= the complete undirected graph K4 with one edge removed) by diagonal inside box sign. A λ-fold 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 λKn with vertex set X. Let (X, B) be a λ-fold block design with block size 4. If we remove one edge from each block in B, we obtain a partial λ-fold diagonal inside box sign -design. If the deleted edges can be arranged into copies of diagonal inside box sign the result is a λ-fold diagonal inside box sign -design, called a metamorphosis of the λ-fold block design (X, B). Quite recently C. C. Lindner and A. Rosa [5] determined the spectrum for λ-fold block designs with block size 4 having a metamorphosis into λ-fold diagonal inside box sign -designs when λ = 1. In this paper we give a complete solution of the spectrum problem for λ-fold block designs with block size 4 having a metamorphosis into a λ-fold diagonal inside box sign -design for all λ ≥ 2.
