The full metamorphosis of λ-fold block designs with block size four into λ-fold triple systems

Abstract

Let (X, B) be a λ-fold block design with block size 4. If a star is removed from each block of B the resulting collection of triangles T is a partial λ-fold triple system (X, T). If the edges belonging to the deleted stars can be arranged into a collection of triangles S, then (X,T ∪ S) is a λ-fold triple system, called a metamorphosis of the A-fold block design (X, B) into a λ-fold triple system. Label the elements of each block b with b 1,b 2,b 3, and b 4 (in any manner). For each i = 1,2,3,4 define a set of triangles T iand a set of stars S i as follows: for each block b = [b 1,b 2,b 3,b 4] belonging to B, partition b into a triangle and a star centered at b i, and place the triangle in T i and the star in S i. Then (X, T i) is a partial λ-fold triple system. Now if the edges belonging to the stars in S i can be arranged into a collection of triangles S i, then (X, T i ∪ S i) is a λ-fold triple system and we say that M i = (X, T i∪S i) is the ith metamorphosis of (X, B). The full metamorphosis of (X, B) is the set of four metamorphoses {M 1, M 2, M 3, M 4}. The purpose of this work is to give a complete solution of the following problem: For which n and λ does there exist a λ-fold block design with block size 4 having a full metamorphosis into λ-fold triple systems? Copyright © 2012, Charles Babbage Research Centre All rights reserved.