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

Abstract

We exhibit cyclic ðK v v ; C k Þ-designs with v v > k; v v k ðmod 2kÞ, for k an odd prime power but not a prime, and for k ¼ 15. Such values were the only ones not to be analyzed yet, under the hypothesis v v k (mod 2kÞ: Our construction avails of Rosa sequences and approximates the Hamiltonian case ðv v ¼ kÞ; 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 ðK v v ; C k Þ-designs with k a prime power. #

Cyclic m‐cycle systems with m ≤ 32 or m = 2q with q a prime power

“…Recently, Buratti and Del Fra [6] present the result that if m is an odd integer with m 6 ¼ 15 and m 6 ¼ p where p is prime and > 1, then there exists a cyclic m-cycle system of order 2km þ m with exception: (m, k) ¼ (3, 1). More recently, Vietri [16] has completely filled in the gap created by Buratti and Del Fra [6]. So we have the following results.…”

“…3, there exists a cyclic m-cycle system of order 2km þ 1. Theorem 1.2 [6,16]. Given an odd integer m !…”

“…With the joint effort of a number of researches [5][6][7][8][9][10][11][12][13]16], the existence question for cyclic q-cycle systems has been settled for q a prime power. When q is not a prime power, the problem becomes much more difficult and is far from being solved.…”

On the Existence of Elementary Abelian Cycle Systems

“…Many authors have contributed to prove the following Theorem 1.1 which is, so far, the most important result about the existence of cyclic k-cycle systems (see [10], [11], [14], [18], [19], [20], [21], [24]). In what follows we will deal with the existence problem of cycle systems for the elementary abelian case.…”

A complete solution to the existence problem for 1‐rotational k‐cycle systems of Kv

“…The existence problem for a cyclic k-cycle system of K v has been solved, with the contribution of many authors ( [10,11,14,16,[18][19][20]23]), in the case of v ≡ 1 or k (mod 2k) (see also [2,4,12]). In all other cases it is conjectured (see [9,24]) that a cyclic k-cycle system of K v exists for any admissible pair (k, v) provided v>2k whenever gcd(k, v) is a prime power.…”