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 ${\mathcal{C}}^{\prime }$ 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 ${\mathcal{D}}_i^{\ast }$, then $M_i=(X,{\mathcal{C}}_i\cup {\mathcal{D}}_i^{\ast })$ 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?