The metamorphosis of λ-fold block designs with block size four into a maximum packing of λKn with 4-cycles

Abstract

Let (X, B) be a lambda-fold block design with block size four and define sets B(C) and E(K-4\C) as follows: for each block b is an element of B, partition b into a 4-cycle and a pair of disjoint edges and place the 4-cycle in B(C) and the 2 disjoint edges in E(K-4\C). If we can reassemble the edges belonging to E(K-4\C) into a collection of 4-cycles E(C) with leave L, then (X, B(C) boolean OR E(C), L) is a packing of A with 4-cycles and is called a metamorphosis of the A-fold block design (X,B). In this paper we give a complete solution of the metamorphosis problem for lambda-fold block designs into maximum packings of lambdaK(n), with 4-cycles for all A (with the possible exception of lambda = 1, n = 37, and leave 2 disjoint triangles). That is, for each lambda we determine the set of all n such that there exists a lambda-fold block design of order n having a metamorphosis into a maximum packing of lambdaK(n) with 4-cycles. (C) 2003 Elsevier B.V. All rights reserved.