$k$-Perfect $3k$-Cycle Systems

Abstract

The spectrum for $k$-perfect $3k$-cycle systems is considered here for arbitrary $k\not\equiv 0(\mathrm{mod}3)$. Previously, the spectrum when $k=2$ was dealt with by Lindner, Phelps, and Rodger, and that for $k=3$ by the current authors. Here, when $k\equiv 1$ or $5(\mathrm{mod}6)$ and $6k+1$ is prime, we show that the spectrum for $k$-perfect $3k$-cycle systems includes all positive integers congruent to $1(\mathrm{mod}6k)$ (except possibly the isolated case $12k+1$). We also complete the spectrum for $k=4$ and $5$ (except possibly for one isolated case when $k=5$), and deal with other specific small values of $k$.