The Metamorphosis of $\lambda$-fold Block Designs with Block Size Four into $\lambda$-fold Kite Systems

Abstract

A kite is a triangle with a tail consisting of a single edge. A kite system of order $n$ is a pair $(X,K)$, where $K$ is a collection of edge disjoint kites which partitions the edge set of $K_n$ (= the complete undirected graph on $n$ vertices) with vertex set $X$. Let $(X,B)$ be a block design with block size 4. If we remove a path of length 2 from each block in $B$, we obtain a partial kite-system. If the deleted edges can be assembled into kites the result is a kite system, called a \emph{metamorphosis} of the block design $(X,B)$. There is an obvious extension of this definition to $\lambda$-fold block designs with block size 4. In this paper we give a complete solution of the following problem: Determine all pairs $(\lambda ,n)$ such that there exists a $\lambda$-fold block design of order $n$ with block size 4 having a metamorphosis into a $\lambda$-fold kite system.