Rotational <i>k</i>‐cycle systems of order <i>v</i> &lt; 3<i>k</i>; another proof of the existence of odd cycle systems

Abstract

Abstract We give an explicit solution to the existence problem for 1‐rotational k ‐cycle systems of order v &lt; 3 k with k odd and v ≠ 2 k + 1. We also exhibit a 2‐rotational k ‐cycle system of order 2 k + 1 for any odd k . Thus, for k odd and any admissible v &lt; 3 k there exists a 2‐rotational k ‐cycle system of order v . This may also be viewed as an alternative proof that the obvious necessary conditions for the existence of odd cycle systems are also sufficient. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 433–441, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10061