Cyclic <i>k</i>‐cycle systems of order 2<i>kn</i> + <i>k</i>: A solution of the last open cases

Abstract

Abstract We exhibit cyclic ( K v , C k )‐designs with v &gt; k , v ≡ k (mod 2 k ), for k an odd prime power but not a prime, and for k = 15. Such values were the only ones not to be analyzed yet, under the hypothesis v ≡ k (mod 2 k ). Our construction avails of Rosa sequences and approximates the Hamiltonian case ( v = k ), which is known to admit no cyclic design with the same values of k . As a particular consequence, we settle the existence question for cyclic ( K v , C k )‐designs with k a prime power. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 299–310, 2004.