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1.
We extend the set of values of n for which it is known that a Z-cyclic triple whist tournament for 4n players exists by proving that if there exists such a tournament for q + 1 players, where q ≡ 3 (mod 4) is prime, then there exists such a tournament for qpa11pann + 1 players, whenever the pi are primes ≡ 5 (mod 8). © 1995 John Wiley & Sons, Inc.  相似文献   

2.
Y. Lu  L. Zhu 《组合设计杂志》1997,5(4):249-256
It is well known that a triplewhist tournament TWh(v) exists only if v ≡ 0 or 1 (mod 4) and v ¬ 5, 9. In this article, we introduce a new concept TWh-frame and use it to show that the necessary condition for the existence of a TWh(v) is also sufficient with a handful possible exceptions of v ∞ {12, 56} ∪ {13, 17, 45, 57, 65, 69, 77, 85, 93, 117, 129, 133, 153}. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 249–256, 1997  相似文献   

3.
A directed triplewhist tournament on v players, briefly DTWh(v), is said to have the three person property if no two games in the tournament have three common players. We briefly denote such a design as a 3PDTWh(v). In this paper, we show that a 3PDTWh(v) exists whenever v>17 and with few possible exceptions.  相似文献   

4.
When the number of players, v, in a whist tournament, Wh(v), is ≡ 1 (mod 4) the only instances of a Z-cyclic triplewhist tournament, TWh(v), that appear in the literature are for v = 21,29,37. In this study we present Z-cyclic TWh(v) for all vT = {v = 8u + 5: v is prime, 3 ≤ u ≤ 249}. Additionally, we establish (1) for all vT there exists a Z-cyclic TWh(vn) for all n ≥ 1, and (2) if viT, i = 1,…,n, there exists a Z-cyclic TWh(v… v) for all ?i ≥ 1. It is believed that these are the first instances of infinite classes of Z-cyclic TWh(v), v ≡ 1 (mod 4). © 1994 John Wiley & Sons, Inc.  相似文献   

5.
The necessary conditions for existence of a triplewhist tournament TWh(v) are . By the efforts of many authors through a century, these conditions are shown to be sufficient except for v=5,9,12,13 and possibly for v=17. A triplewhist tournament Wh(v) is said to have the three person property if any two games in the tournament do not have three common players. We briefly denote such a design as a 3PTWh(v). In this paper, we extend the known existence result for TWh(v)s and show that the necessary conditions for existence of a 3PTWh(v), namely, v?8 and , are also sufficient except for v=9,12,13 and possibly for v=17.  相似文献   

6.
An RTD[5,λ; v] is a decomposition of the complete symmetric directed multigraph, denoted by λK, into regular tournaments of order 5. In this article we show that an RTD[5,λ; v] exists if and only if (v?1)λ ≡ 0 (mod 2) and v(v?1)λ ≡ 0 (mod 10), except for the impossible case (v,λ) = (15,1). Furthermore, we show that for each v ≡ 1,5 (mod 20), v ≠ 5, there exists a B[5,2; v] which is not RT5-directable. © 1994 John Wiley & Sons, Inc.  相似文献   

7.
A directed triplewhist tournament on p players over Z p is said to have the three-person property if no two games in the tournament have three common players. We briefly denote such a design as a 3PDTWh(p). In this paper, we investigate the existence of a Z-cyclic 3PDTWh(p) for any prime p ≡ 1 (mod 4) and show that such a design exists whenever p ≡ 5, 9, 13 (mod 16) and p ≥ 29. This result is obtained by applying Weil’s theorem. In addition, we also prove that a Z-cyclic 3PDTWh(p) exists whenever p ≡ 1 (mod 16) and p < 10, 000 except possibly for p = 257, 769. Gennian Ge’s Research was supported by National Natural Science Foundation of China under Grant No. 10471127, Zhejiang Provincial Natural Science Foundation of China under Grant No. R604001, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.  相似文献   

8.
It is shown that for each integer m ≥ 1 there exists a lower bound, vm, with the property that for all vvm with v ≡ 1, 4 (mod 12) there exists an m-chromatic S(2, 4, v) design. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 403–409, 1998  相似文献   

9.
In this article, we construct LGDD(mv) for v ≡ 2 (mod 6) and m ≡ 0 (mod 6) by constructing the missing essential values 6v, v = 8, 14, 26, 50. Thus an LGDD(mv) exists if and only if v(v - 1)m2 ≡ 0 (mod 6), (v - 1)m ≡ 0 (mod 2) and (m, v) ≠ (1, 7). © 1997 John Wiley & Sons, Inc.  相似文献   

10.
We show that a Z-cyclic triplewhist tournament on p players with three-person property, briefly 3PTWh(p), exists for any prime p≡ 1 (mod 4) with the only exceptions of p=5,13,17.  相似文献   

11.
A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, Cλ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: XZm such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ?1(c)∣ cZm differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5a 13b 17c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5a 13b 17c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.  相似文献   

12.
A quadruple system of order v, denoted QS(v) is an ordered pair (X, Q) where X is a set of cardinality v and Q is a set of 4-subsets of X called blocks, with the property that every 3-subset of X is contained in a unique block. The points and planes of the affine geometry AG(3, 2) form a QS(8). We prove that a QS(v) containing a proper subsystem isomorphic to AG(3, 2) exists if and only if v?16 and v≡2 or 4 (mod 6).  相似文献   

13.
In this article, we construct pairwise balanced designs (PBDs) on v points having blocks of size five, except for one block of size w ? {17,21,25,29,33}. A necessary condition for the existence of such a PBD is v ? 4w + 1 and (1) v ≡ 1 or 5 (mod 20) for w = 21, 25; (2) v ≡ 9 or 17 (mod 20) for w = 17,29; (3) v ≡ 13 (mod 20) for w = 33. We show that these necessary conditions are sufficient with at most 25 possible exceptions of (v,w). We also show that a BIBD B(5, 1; w) can be embedded in some B(5, 1; v) whenever vw ≡ 1 or 5 (mod 20) and v ? 5w ? 4, except possibly for (v, w) = (425, 65). © 1995 John Wiley & Sons, Inc.  相似文献   

14.
The obvious necessary conditions for the existence of a nested Steiner triple system of order v containing a nested subsystem of order w are v ≥ 3w + 4 and v ≡ w ≡ 1 (mod 6). We show that these conditions are also sufficient. © 2004 Wiley Periodicals, Inc.  相似文献   

15.
It is well known that every tournament contains a Hamiltonian path, which can be restated as that every tournament contains a unary spanning tree. The purpose of this article is to study the general problem of whether a tournament contains a k‐ary spanning tree. In particular, we prove that, for any fixed positive integer k, there exists a minimum number h(k) such that every tournament of order at least h(k) contains a k‐ary spanning tree. The existence of a Hamiltonian path for any tournament is the same as h(1) = 1. We then show that h(2) = 4 and h(3) = 8. The values of h(k) remain unknown for k ≥ 4. © 1999 John & Sons, Inc. J Graph Theory 30: 167–176, 1999  相似文献   

16.
The spectrum for LMTS(v,1) has been obtained by Kang and Lei (Bulletin of the ICA, 1993). In this article, firstly, we give the spectrum for LMTS(v,3). Furthermore, by the existence of LMTS(v,1) and LMTS(v,3), the spectrum for LMTS(v,λ) is completed, that is v ≡ 2 (mod λ), v ≥ λ + 2, if λ ? 0(mod 3) then v ? 2 (mod 3) and if λ = 1 then v ≠ 6. © 1994 John Wiley & Sons, Inc.  相似文献   

17.
A tournament is an orientation of the edges of a complete graph. An arc is pancyclic in a tournament T if it is contained in a cycle of length l, for every 3 ≤ l ≤ |T|. Let p(T) denote the number of pancyclic arcs in a tournament T. In 4 , Moon showed that for every non‐trivial strong tournament T, p(T) ≥ 3. Actually, he proved a somewhat stronger result: for any non‐trivial strong tournament h(T) ≥ 3 where h(T) is the maximum number of pancyclic arcs contained in the same hamiltonian cycle of T. Moreover, Moon characterized the tournaments with h(T) = 3. All these tournaments are not 2‐strong. In this paper, we investigate relationship between the functions p(T) and h(T) and the connectivity of the tournament T. Let pk(n) := min {p(T), T k‐strong tournament of order n} and hk(n) := min{h(T), T k‐strong tournament of order n}. We conjecture that (for k ≥ 2) there exists a constant αk> 0 such that pk(n) ≥ αkn and hk(n) ≥ 2k+1. In this paper, we establish the later conjecture when k = 2. We then characterized the tournaments with h(T) = 4 and those with p(T) = 4. We also prove that for k ≥ 2, pk(n) ≥ 2k+3. At last, we characterize the tournaments having exactly five pancyclic arcs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 87–110, 2004  相似文献   

18.
It is shown that for any even integer n ≥ 16 and for some b ≥ 1, there exists a skew Room frame of type 2a 4b with n = 2a + 4b. As an application, we show that for any integer v ≡ 1,4 (mod 12), there exists a weakly 3-chromatic linear space with all lines having size 4 iff v > 16. This answers an open problem posed by Colbourn and Rosa. © 1994 John Wiley & Sons, Inc.  相似文献   

19.
We construct simple indecomposable twofold cyclic triple systems TS2(v) for all v ≡ 0, 1, 3, 4, 7, and 9(mod 12), where v = 4 or v ≥ 12, using Skolem‐type sequences. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 402–410, 2000  相似文献   

20.
L. Ji 《组合设计杂志》2004,12(2):92-102
Let B3(K) = {v:? an S(3,K,v)}. For K = {4} or {4,6}, B3(K) has been determined by Hanani, and for K = {4, 5} by a previous paper of the author. In this paper, we investigate the case of K = {4,5,6}. It is easy to see that if vB3 ({4, 5, 6}), then v ≡ 0, 1, 2 (mod 4). It is known that B3{4, 6}) = {v > 0: v ≡ 0 (mod 2)} ? B3({4,5,6}) by Hanani and that B3({4, 5}) = {v > 0: v ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v ≠ 13} ? B3({4, 5, 6}). We shall focus on the case of v ≡ 9 (mod 12). It is proved that B3({4,5,6}) = {v > 0: v ≡ 0, 1, 2 (mod 4) and v ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.  相似文献   

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