A family (X, B1), (X, B2), . . . , (X, Bq) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTSλ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and d...A family (X, B1), (X, B2), . . . , (X, Bq) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTSλ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTSλ(v) if there exists no LSTSλ (v) contained in the collection for any λ 【 λ. In 1995, Griggs and Rosa posed a problem: For which values of λ 】 1 and orders v ≡ 1, 3 (mod 6) do there exist IDLSTSλ(v)? In this paper, we use partitionable candelabra systems (PCSs) and holey λ-fold large set of STS(v) (HLSTSλ(v)) as auxiliary designs to establish a recursive construction for IDLSTSλ(v) and show that there exists an IDLSTSλ(v) for λ = 2, 3, 4 and v ≡ 1, 3 (mod 6).展开更多
A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposab...A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposable and denoted by IDLSTSx（v） if there does not exist an LSTSx, （v） contained in the collection for any λ 〈 λ. In this paper, we show that for λ = 5, 6, there is an IDLSTSλ（v） for v ≡ 1 or 3 （rood 6） with the exception IDLSTS6（7）.展开更多
An LPDTS(υ) is a collection of 3(υ - 2) disjoint pure directed triple systems on the same set of υ elements. It is showed in Tian's doctoral thesis that there exists an LPDTS(υ) for υ≡ 0,4 (mod 6), υ≥ 4. I...An LPDTS(υ) is a collection of 3(υ - 2) disjoint pure directed triple systems on the same set of υ elements. It is showed in Tian's doctoral thesis that there exists an LPDTS(υ) for υ≡ 0,4 (mod 6), υ≥ 4. In this paper, we establish the existence of an LPDTS(υ) for υ≡ 1, 3 (mod 6), υ＞ 3. Thus the spectrum for LPDTS(υ) is completely determined to be the set {υ:υ≡0, 1 (mod 3),υ≥4}.展开更多
In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a trip...In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang （J. Combin. Designs, 4 （1996）, 301-321）. As an application, we obtain an LRDTS（4·3^n） for any integer n ≥ 1, which provides an infinite family of even orders.展开更多
A Mendelsohn triple system of order v （MTS（v）） is a pair （X,B） where X is a v-set and 5g is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of B. An MTS（v） ...A Mendelsohn triple system of order v （MTS（v）） is a pair （X,B） where X is a v-set and 5g is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of B. An MTS（v） （X,B） is called pure and denoted by PMTS（v） if （x, y, z） ∈ B implies （z, y, x） ∈B. A large set of MTS（v）s （LMTS（v）） is a collection of v - 2 pairwise disjoint MTS（v）s on a v-set. A self-converse large set of PMTS（v）s, denoted by LPMTS＊ （v）, is an LMTS（v） containing [ v-2/2] converse pairs of PMTS（v）s. In this paper, some results about the existence and non-existence for LPMTS＊ （v） are obtained.展开更多
In this paper, we introduce a new concept -- overlarge sets of generalized Kirkman systems （OLGKS）, research the relation between it and OLKTS, and obtain some new results for OLKTS. The main conclusion is： If ther...In this paper, we introduce a new concept -- overlarge sets of generalized Kirkman systems （OLGKS）, research the relation between it and OLKTS, and obtain some new results for OLKTS. The main conclusion is： If there exist both an OLKF（6^k） and a 3-OLGKS（6^k-1,4） for all k ∈{6,7,...,40}／{8,17,21,22,25,26}, then there exists an OLKTS（v） for any v ≡ 3 （mod 6）, v ≠ 21. As well, we obtain the following result： There exists an OLKTS（6u ＋ 3） for u = 2^2n-1 - 1, 7^n, 31^n, 127^n, 4^r25^s, where n ≥ 1,r＋s≥ 1.展开更多
A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to...A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS（v, λ） is a DTS（v, λ） without repeated blocks. A simple DTS（v, ）,） is called pure and denoted by PDTS（v, λ） if （x, y, z） ∈ B implies （z, y, x）, （z, x, y）, （y, x, z）, （y, z, x）, （x, z, y） B. A large set of disjoint PDTS（v, λ）, denoted by LPDTS（v, λ）, is a collection of 3（v - 2）/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS（v, λ） are presented. Especially, we determine the spectrum of LPDTS（v, 2）.展开更多
A hybrid triple system of order v, briefly by HTS （v）, is a pair （X,/3） where X is a v-set and /3 is a collection of cyclic and transitive triples （called blocks） on X such that every ordered pair of X belongs t...A hybrid triple system of order v, briefly by HTS （v）, is a pair （X,/3） where X is a v-set and /3 is a collection of cyclic and transitive triples （called blocks） on X such that every ordered pair of X belongs to exactly one block of/3. An HTS （v） is called pure and denoted by PHTS （v） if one element of the block set {（x,y,z）, （z,y, ss）, （z,sc,y）, （y,x,z）, （y,z,x）, （x,z,y）, （x,y,z）, （z,y,x）} is contained in 13 then the others will not be contained in/3. A self-converse large set of disjoint PHTS （v）s, denoted by LPHTS＊（v）, is a collection of 4（v - 2） disjoint PHTS （v）s which contains exactly （v - 2）/2 converse octads of PHTS （v）s. In this paper, some results about the existence for LPHTS＊ （v） are obtained.展开更多
In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a...In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a computer, we find examples of such structure for t C T = {0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 20, 22, 24}. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for LR-design in [8], we obtain the existence for LRMTS（v）and LRDTS（v）, where v = 12（t ＋ 1） mi≥0（2.7mi＋1）mi≥0（2.13ni＋1）and t∈T,which provides more infinite family for LRMTS and LRDTS of even orders.展开更多
An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order ...An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X,B), where X is an nelement set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X, B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n≡2,4(mod 6),n＞4,or n≡1,5(mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n≡2,4 (mod 6) and n＞4, or n≡1, 5 (mod 12).Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n≡1,3(mod 6),n＞3, or n≡0,4 (mod 12).展开更多
In this article, we establish the existence of an LHMTS（mv） for v ≡ 2 （mod 6） and m≡ 3 （mod 6）. Thus there exists an LHMTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3） except possibly for v=6, m≡ 1, 5 （mo...In this article, we establish the existence of an LHMTS（mv） for v ≡ 2 （mod 6） and m≡ 3 （mod 6）. Thus there exists an LHMTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3） except possibly for v=6, m≡ 1, 5 （mod 6） and m≠1. In the similar way, the existence of LHDTS（mv） is completely determined, i.e., there exists an LHDTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3）.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.10971051, 10701060, 10831002)Qing Lan Project of Jiangsu Province, China
文摘A family (X, B1), (X, B2), . . . , (X, Bq) of q STS(v)s is a λ-fold large set of STS(v) and denoted by LSTSλ(v) if every 3-subset of X is contained in exactly λ STS(v)s of the collection. It is indecomposable and denoted by IDLSTSλ(v) if there exists no LSTSλ (v) contained in the collection for any λ 【 λ. In 1995, Griggs and Rosa posed a problem: For which values of λ 】 1 and orders v ≡ 1, 3 (mod 6) do there exist IDLSTSλ(v)? In this paper, we use partitionable candelabra systems (PCSs) and holey λ-fold large set of STS(v) (HLSTSλ(v)) as auxiliary designs to establish a recursive construction for IDLSTSλ(v) and show that there exists an IDLSTSλ(v) for λ = 2, 3, 4 and v ≡ 1, 3 (mod 6).
基金Supported by National Natural Science Foundation of China (Grant Nos. 10971051 and 11071056)
文摘A family （X, B1）,（X, B2）,..., （X, Bq） of q STS（v）s is a λ-fold large set of STS（v） and denoted by LSTSλ（v） if every 3-subset of X is contained in exactly A STS（v）s of the collection. It is indecomposable and denoted by IDLSTSx（v） if there does not exist an LSTSx, （v） contained in the collection for any λ 〈 λ. In this paper, we show that for λ = 5, 6, there is an IDLSTSλ（v） for v ≡ 1 or 3 （rood 6） with the exception IDLSTS6（7）.
基金This work was partially supported by the National Natural Science Foundation of China(Grant No.10371002)the Tianyuan Mathematics Foundation of the National Natural Science Foundation of China(Grant No.10526032)the Natural Science Foundation of Universities of Jiangsu Province(Grant No.05KJB110111).
文摘An LPDTS(υ) is a collection of 3(υ - 2) disjoint pure directed triple systems on the same set of υ elements. It is showed in Tian's doctoral thesis that there exists an LPDTS(υ) for υ≡ 0,4 (mod 6), υ≥ 4. In this paper, we establish the existence of an LPDTS(υ) for υ≡ 1, 3 (mod 6), υ＞ 3. Thus the spectrum for LPDTS(υ) is completely determined to be the set {υ:υ≡0, 1 (mod 3),υ≥4}.
文摘In this paper, we first define a doubly transitive resolvable idempotent quasigroup （DTRIQ）, and show that aDTRIQ of order v exists if and only ifv ≡0（mod3） and v ≠ 2（mod4）. Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang （J. Combin. Designs, 4 （1996）, 301-321）. As an application, we obtain an LRDTS（4·3^n） for any integer n ≥ 1, which provides an infinite family of even orders.
基金Supported by National Natural Science Foundation of China (Grant No.10771051)
文摘A Mendelsohn triple system of order v （MTS（v）） is a pair （X,B） where X is a v-set and 5g is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of B. An MTS（v） （X,B） is called pure and denoted by PMTS（v） if （x, y, z） ∈ B implies （z, y, x） ∈B. A large set of MTS（v）s （LMTS（v）） is a collection of v - 2 pairwise disjoint MTS（v）s on a v-set. A self-converse large set of PMTS（v）s, denoted by LPMTS＊ （v）, is an LMTS（v） containing [ v-2/2] converse pairs of PMTS（v）s. In this paper, some results about the existence and non-existence for LPMTS＊ （v） are obtained.
基金supported by NSFC Grant 10671055NSFHB A2007000230Foundation of Hebei Normal University L2004Y11, L2007B22
文摘In this paper, we introduce a new concept -- overlarge sets of generalized Kirkman systems （OLGKS）, research the relation between it and OLKTS, and obtain some new results for OLKTS. The main conclusion is： If there exist both an OLKF（6^k） and a 3-OLGKS（6^k-1,4） for all k ∈{6,7,...,40}／{8,17,21,22,25,26}, then there exists an OLKTS（v） for any v ≡ 3 （mod 6）, v ≠ 21. As well, we obtain the following result： There exists an OLKTS（6u ＋ 3） for u = 2^2n-1 - 1, 7^n, 31^n, 127^n, 4^r25^s, where n ≥ 1,r＋s≥ 1.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10771013 and 10831002)
文摘A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS（v, λ） is a DTS（v, λ） without repeated blocks. A simple DTS（v, ）,） is called pure and denoted by PDTS（v, λ） if （x, y, z） ∈ B implies （z, y, x）, （z, x, y）, （y, x, z）, （y, z, x）, （x, z, y） B. A large set of disjoint PDTS（v, λ）, denoted by LPDTS（v, λ）, is a collection of 3（v - 2）/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS（v, λ） are presented. Especially, we determine the spectrum of LPDTS（v, 2）.
基金Supported by Tianyuan Mathematics Foundation of National Natural Science Foundation of China(No.11126285)
文摘A hybrid triple system of order v, briefly by HTS （v）, is a pair （X,/3） where X is a v-set and /3 is a collection of cyclic and transitive triples （called blocks） on X such that every ordered pair of X belongs to exactly one block of/3. An HTS （v） is called pure and denoted by PHTS （v） if one element of the block set {（x,y,z）, （z,y, ss）, （z,sc,y）, （y,x,z）, （y,z,x）, （x,z,y）, （x,y,z）, （z,y,x）} is contained in 13 then the others will not be contained in/3. A self-converse large set of disjoint PHTS （v）s, denoted by LPHTS＊（v）, is a collection of 4（v - 2） disjoint PHTS （v）s which contains exactly （v - 2）/2 converse octads of PHTS （v）s. In this paper, some results about the existence for LPHTS＊ （v） are obtained.
基金the National Natural Science Foundation of China(No.10671055)Natural Science Foundation of Hebei(No.A2007000230)
文摘In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a computer, we find examples of such structure for t C T = {0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 20, 22, 24}. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for LR-design in [8], we obtain the existence for LRMTS（v）and LRDTS（v）, where v = 12（t ＋ 1） mi≥0（2.7mi＋1）mi≥0（2.13ni＋1）and t∈T,which provides more infinite family for LRMTS and LRDTS of even orders.
基金This work was partially supported by the Tianyuan Mathematics Foundation of NSFC(Grant No.10526032)the Natural Science Foundation of Universities of Jiangsu Province(Grant No.05KJB110111).
文摘An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X,B), where X is an nelement set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X, B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n≡2,4(mod 6),n＞4,or n≡1,5(mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n≡2,4 (mod 6) and n＞4, or n≡1, 5 (mod 12).Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n≡1,3(mod 6),n＞3, or n≡0,4 (mod 12).
基金Supported by National Natural Science Foundation of China(Grant Nos.11471096 and 11771119)
文摘In this article, we establish the existence of an LHMTS（mv） for v ≡ 2 （mod 6） and m≡ 3 （mod 6）. Thus there exists an LHMTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3） except possibly for v=6, m≡ 1, 5 （mod 6） and m≠1. In the similar way, the existence of LHDTS（mv） is completely determined, i.e., there exists an LHDTS（mv） if and only if v（v-1）m2 ≡ 0 （mod 3）.
