The Full Metamorphosis of \(\lambda\)-fold Block Designs with Block Size Four into \(\lambda\)-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 \(\mathcal{B}\), the resulting collection of \(4\)-cycles \(\mathcal{C}’\) is a partial \(\lambda\)-fold \(4\)-cycle system \((X, \mathcal{C})\). If the deleted edges can be arranged into a collection of \(4\)-cycles \(\mathcal{D}\), then \((X, \mathcal{C} \cup \mathcal{D})\) is a \(\lambda\)-fold \(4\)-cycle system [10]. Now for each block \(b \in {B}\), 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 \(\mathcal{C}_i\) and \(\mathcal{D}_i\) as follows: for each \(b \in {B}\), put the \(4\)-cycle \(b \setminus F_i(b)\) in \(\mathcal{C}_i\) and the \(2\) edges belonging to \(F_i(b)\) in \(\mathcal{D}_i\). If the edges in \(\mathcal{D}_i\) can be arranged into a collection of \(4\)-cycles \(\mathcal{D}^*_i\), then \( {M}_i = (X, \mathcal{C}_i \cup \mathcal{D}^*_i)\) is a \(\lambda\)-fold 4-cycle system, called the \(i\)th metamorphosis of \((X, \mathcal{B})\). 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?