The full metamorphosis of lambda-fold block designs with block size four into lambda-fold kite systems

Abstract

Let(X,B)be a λ-fold block design with block size 4. If a path of length two is removed from each block of B the resulting collection of kites K is a partial λ-fold kite system(X,K). If the deleted edges can be arranged into a collection of kites D,then(X,K ∪ D)is a λ-fold kite system [5]. Now for each block 6 ∈ B let {P1(6),P 2(b),P3(b)} be a partition of b into paths of length two and define for each i = 1,2,3, sets Ki and Di as follows: for each b ∈ B,put the kite b\Pi(b)in Ki and the two edges belonging to the path Pi(b)in Di. If the edges in Di can be arranged into a collection of kites Di * then Mi =(X,Ki∪Di *)is a λ-fold kite system,called the ith metamorphosis of(X,B). The full metamorphosis is the set of three metamorphoses {M 1,M2,M3}. We give a complete solution of the following problem: for which n and A does there exist a λ-fold block design with block size 4 having a full metamorphosis into a λ-fold kite system?