The full metamorphosis of λ-fold block designs with block size four into A-fold 4-cycle systems

Abstract

Let (X, B) be a lambda-fold block design with block size 4. If a pair of disjoint edges are removed from each block of B the resulting collection of 4-cycles C is a partial lambda-fold 4-cycle system (X, C). If the deleted edges can be arranged into a collection of 4-cycles D, then (X,C boolean OR D) is a lambda-fold 4-cycle system [10]. Now for each block b is an element of B specify a 1-factorization of b as {F-1(b), F-2(b), F-3(b)} and define for each i = 1,2,3, sets C-i and D-i as follows: for each b is an element of B, put the 4-cycle b backslash F-i(b) in C-i and the 2 edges belonging to F-i(b) in D-i. If the edges in D-i can be arranged into a collection of 4-cycles D-i* then M-i = (X, C-i boolean OR D-i*) is a lambda-fold 4-cycle system, called the ith metamorphosis of (X, B). The full metamorphosis is the set of three metamorphoses {M-1, M-2, M-3}. We give a complete solution of the following problem: for which n and lambda does there exist a lambda-fold block design with block size 4 having a full metamorphosis into a lambda-fold 4-cycle system?