3-Simple 2-Fold 5-Cycle Systems

Abstract

A decomposition of $\lambda K_n$ into cycles of length k is called a $\lambda$-fold k-cycle system of $\lambda K_n$. A $\lambda$-fold k-cycle system of $\lambda K_n$ is t-simple, $t<k$, if any two cycles in the decomposition have at most t vertices in common. We denote a t-simple $\lambda$-fold k-cycle system of $\lambda K_n$ by $(n,k,\lambda ,t)$-cycle system. In this paper, it is shown that an (n, 5, 2, 3)-cycle system exists, for $n=5r,\ 5r+1$ when (i) $r\equiv$ 2 or 6 (mod 12) or (ii) $r\equiv$ 4 or 12 (mod 24).