The Metamorphosis of Lambda-Fold $K_{3,3}$-Designs into Lambda-Fold $6$-Cycle Systems

Abstract

In this paper, necessary and sufficient conditions are given for the metamorphosis of a $\lambda$-fold $K_{3,3}$-design of order $n$ into a $\lambda$-fold $6$-cycle system of order $n$, by retaining one $6$-cycle subgraph from each copy of $K_{3,3}$, and then rearranging the set of all the remaining edges, three from each $K_{3,3}$, into further $6$-cycles so that the result is a $\lambda$-fold $6$-cycle system.