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In this article, it is shown that there exists a 1‐rotationally resolvable 4‐cycle system of 2Kυ if and only if υ ≡ 0 (mod 4). To prove that, some special sequences of integers are utilized. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 116–125, 2002; DOI 10.1002/jcd.10006 相似文献
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Fu and Mishima [J. Combin. Des. 10 (2002), pp. 116–125] have utilized the extended Skolem sequence to prove that there exists a 1‐rotationally resolvable ‐cycle system of if and only if (mod ). In this paper, the existence of a cyclically near‐resolvable ‐cycle system is discussed, and it is shown that there exists a cyclically near‐resolvable ‐cycle system of if and only if (mod ). 相似文献
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The necessary and sufficient conditions for the existence of a 1‐rotational k‐cycle system of the complete graph Kv are established. The proof provides an algorithm able to determine, directly and explicitly, an odd k‐cycle system of Kv whenever such a system exists. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 283–293, 2009 相似文献
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Marco Buratti 《组合设计杂志》2001,9(3):215-226
Phelps and Rosa introduced the concept of 1‐rotational Steiner triple system, that is an STS(ν) admitting an automorphism consisting of a fixed point and a single cycle of length ν ? 1 [Discrete Math. 33 ( 12 ), 57–66]. They proved that such an STS(ν) exists if and only if ν ≡ 3 or 9 (mod 24). Here, we speak of a 1‐rotational STS(ν) in a more general sense. An STS(ν) is 1‐rotational over a group G when it admits G as an automorphism group, fixing one point and acting regularly on the other points. Thus the STS(ν)'s by Phelps and Rosa are 1‐rotational over the cyclic group. We denote by ??1r, ??1r, ??1r, ??1r, the spectrum of values of ν for which there exists a 1‐rotational STS(ν) over an abelian, a cyclic, a dicyclic, and an arbitrary group, respectively. In this paper, we determine ??1r and find partial answers about ??1r and ??1r. The smallest 1‐rotational STSs have orders 9, 19, 25 and are unique up to isomorphism. In particular, the only 1‐rotational STS(25) is over SL2(3), the special linear group of dimension 2 over Z3. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 215–226, 2001 相似文献
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It is shown that for any finite group Γ, there exists a 2k‐cycle system whose full automorphism group is isomorphic to Γ. Furthermore, the minimal order of such a system is at most , where . 相似文献
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A group divisible design GD(k,λ,t;tu) is α‐resolvable if its blocks can be partitioned into classes such that each point of the design occurs in precisely α blocks in each class. The necessary conditions for the existence of such a design are λt(u ? 1) = r(k ? 1), bk = rtu, k|αtu and α|r. It is shown in this paper that these conditions are also sufficient when k = 3, with some definite exceptions. © 2004 Wiley Periodicals, Inc. 相似文献
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In this paper, the necessary and sufficient conditions for the existence of cyclic 2q‐cycle and m‐cycle systems of the complete graph with q a prime power and 3 ≤ m ≤ 32 are given. © 2005 Wiley Periodicals, Inc. J Combin Designs 相似文献
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Peter Jenkins 《组合设计杂志》2006,14(4):324-332
In this paper, it is shown that for any pair of integers (m,n) with 4 ≤ m ≤ n, if there exists an m‐cycle system of order n, then there exists an irreducible 2‐fold m‐cycle system of order n, except when (m,n) = (5,5). A similar result has already been established for the case of 3‐cycles. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 324–332, 2006 相似文献
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In this paper, we construct almost resolvable cycle systems of order for odd . This completes the proof of the existence of almost resolvable cycle systems with odd cycle length. As a by-product, some new solutions to the Hamilton–Waterloo problem are also obtained. 相似文献
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Andrea Vietri 《组合设计杂志》2004,12(4):299-310
We exhibit cyclic (Kv, Ck)‐designs with v > k, v ≡ k (mod 2k), 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 2k). 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 (Kv, Ck)‐designs with k a prime power. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 299–310, 2004. 相似文献
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We collect some old and new results on Hamiltonian cycle systems of the complete graph (or the complete graph minus a 1-factor) having an automorphism group that satisfies specific properties. 相似文献
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Let n≥2 be an integer. The complete graph Kn with a 1‐factor F removed has a decomposition into Hamilton cycles if and only if n is even. We show that Kn−F has a decomposition into Hamilton cycles which are symmetric with respect to the 1‐factor F if and only if n≡2, 4 mod 8. We also show that the complete bipartite graph Kn, n has a symmetric Hamilton cycle decomposition if and only if n is even, and that if F is a 1‐factor of Kn, n, then Kn, n−F has a symmetric Hamilton cycle decomposition if and only if n is odd. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:1‐15, 2010 相似文献
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A k‐cycle system of a multigraph G is an ordered pair (V, C) where V is the vertex set of G and C is a set of k‐cycles, the edges of which partition the edges of G. A k‐cycle system of is known as a λ‐fold k‐cycle system of order V. A k‐cycle system of (V, C) is said to be enclosed in a k‐cycle system of if and . We settle the difficult enclosing problem for λ‐fold 5‐cycle systems with u = 1. 相似文献
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Miwako Mishima 《Discrete Mathematics》2008,308(12):2617-2619
The spectrum of values v for which a 1-rotational Steiner triple system of order v exists over a dicyclic group is determined. 相似文献
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A 2‐class regular partial Steiner triple system is a partial Steiner triple system whose points can be partitioned into 2‐classes such that no triple is contained in either class and any two points belonging to the same class are contained in the same number of triples. It is uniform if the two classes have the same size. We provide necessary and sufficient conditions for the existence of uniform 2‐class regular partial Steiner triple systems. 相似文献
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Suppose H is a complete m-partite graph Km(n1,n2,…,nm) with vertex set V and m independent sets G1,G2,…,Gm of n1,n2,…,nm vertices respectively. Let G={G1,G2,…,Gm}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V?Gi for some Gi∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type gu. In this paper, we show that there exists a (k,λ)-cycle frame of type gu for k∈{4,5,6} if and only if , , u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when . We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions. 相似文献
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Marco Buratti 《组合设计杂志》2003,11(6):433-441
We give an explicit solution to the existence problem for 1‐rotational k‐cycle systems of order v < 3k with k odd and v ≠ 2k + 1. We also exhibit a 2‐rotational k‐cycle system of order 2k + 1 for any odd k. Thus, for k odd and any admissible v < 3k 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 相似文献
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An m‐cycle system of order n is a partition of the edges of the complete graph Kn into m‐cycles. We investigate k‐colorings of 4‐cycle systems in which no 4‐cycle is monochromatic. For any k ≥ 3, we construct a k‐chromatic 4‐cycle system. We also show that for any k ge; 2, there exists an integer wk such that for all admissible n ≥ wk, there is a k‐chromatic 4‐cycle system of order n. © 2005 Wiley Periodicals, Inc. J Combin Designs 相似文献