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A completion of the spectrum for large sets of disjoint transitive triple systems

Qingde Kang, Yanxun Chang
1992 Journal of combinatorial theory. Series A  
Thereby, we completely solve the existence problem of LTTS(V) (large set of pairwise disjoint transitive triple systems of order 0).  ...  A SYMMETRIC LTTS(24 + 2) In [4, 51 we introduced a kind of special large set of disjoint Mendelsohn triple systems (LMTS)-symmetric LMTS.  ...  For any v-0 or 1 (mod 3), theve exists an LTTS(u). Proof: Let v = 2% + 2, where u = 1 or 5 (mod 6). If n = 0, by (R2), there exists an LTTS(u).  ... 
doi:10.1016/0097-3165(92)90009-j fatcat:z4p5suncmzasbp7ywzsqpwxeda

Construction of large sets of pairwise disjoint transitive triple systems II

C.C Lindner
1987 Discrete Mathematics  
The maximum number of pairwise disjoint transitive triple systems (T'I'Ss) of order n is 3(n -2). Such a collection is called a large set of pairwise disjoint TI'Ss of order n.  ...  Two consequences of this result are the existence of a large set of pairwise disjoint TI'Ss of every odd admissible order and the existence of a large set of pairwise disjoint TI'Ss of every admissible  ...  Construction of large sets of pairwise disjoint transitive triple systems II 73 Corollary 5.2.  ... 
doi:10.1016/0012-365x(87)90211-1 fatcat:nxpi4w2xlvfnxhcrqlhzyo56je

Page 4708 of Mathematical Reviews Vol. , Issue 93i [page]

1993 Mathematical Reviews  
Phelps (Auburn University, AL) 93i:05024 05B07 Kang, Qing De (PRC-HBN); Chang, Yan Xun (PRC-HBN) A completion of the spectrum for large sets of disjoint transitive triple systems. J. Combin.  ...  A transitive triple system (TTS) of order n is a pair (P,T), where T is a collection of edge disjoint transitive triples which partition the edge set of D,, with vertex set P.  ... 

Page 5806 of Mathematical Reviews Vol. , Issue 96j [page]

1996 Mathematical Reviews  
Math. 27 (1975), 256-260; MR 51 #5334] constructed for every n = 1 or 3 (mod 6) a large set of n triple systems of order n which were mutually almost disjoint (MAD) ((S, T;) and (S,7;) are almost disjoint  ...  It is not hard to see that a large set of MAD triple systems of order n must consist of n—1, n, or n+1 triple systems.  ... 

A class of 2-chromatic SQS(22)

K.T. Phelps
1991 Discrete Mathematics  
We enumerate all 2-chromatic Steiner quadruple systems of order 22, having a cyclic automorphism of order 11 (i.e. 2 disjoint ll-cycles). 0012-365X/91/$03.50 0 1991-Elsevier Science Publishers B.V.  ...  In memory of Egmont Kiihler. Abstract Phelps, K.T., A class of 2khromatic SQS(22), Discrete Mathematics 97 (1991) 333-338.  ...  of triples is a partial triple system of order II, then this will give us the base blocks for a large set of partial triple systems, which we will call a cyclic large set.  ... 
doi:10.1016/0012-365x(91)90448-b fatcat:pjvk6cbgevhmvnkwc6zfmsk2am

Page 2562 of Mathematical Reviews Vol. , Issue 95e [page]

1995 Mathematical Reviews  
spectrum for large sets of disjoint Mendelsohn triple systems with any index.  ...  It is well known that the spectrum (= set of all m such that a triple system of order n exists) for triple systems is precisely the set of all »=1 or 3 (mod 6).  ... 

Page 4764 of Mathematical Reviews Vol. , Issue 98H [page]

1998 Mathematical Reviews  
A large set of mutually almost disjoint (MAD) Steiner triple systems of order v is a collection of Steiner triple systems on the same v-set S such that every 3-subset of S occurs as a block in at least  ...  It is well known that the spectrum (= set of all v for which a /-fold directed triple system of order v exists) for 3-fold directed triple systems consists of all v > 3.  ... 

Page 2266 of Mathematical Reviews Vol. , Issue 85f [page]

1985 Mathematical Reviews  
It is conjectured that a CSQS(v) exists for all admissible orders v except 8, 14 and 16. However, the spectrum for CSQS(v) has not been completely determined.  ...  An invariant for Steiner triple systems (STS) is a mapping f defined on the set of all STSs such that f(D) = f(D’) if D, D’  ... 

Page 8081 of Mathematical Reviews Vol. , Issue 2003k [page]

2003 Mathematical Reviews  
[Zhu, Lie'] (PRC-SOO; Suzhou) An improved product construction for large sets of Kirkman triple systems. (English summary) Discrete Math. 260 (2003), no. 1-3, 307-313.  ...  (v) is a transitive Kirkman triple system of order v, and an LR(uw) is a new kind of design introduced by Lei.  ... 

Page 4059 of Mathematical Reviews Vol. , Issue 97G [page]

1997 Mathematical Reviews  
A large set of disjoint resolv- able Mendelson [directed] triple systems of order v, LRMTS(v) {[LRDTS(v)], is a partition of the set of all cyclic [transitive] triples on a v-set into resolvable Mendelsohn  ...  A directed triple system of or- der v is a decomposition of the complete directed graph, with- out loops, on a v-set into transitive triples, where a transitive triple is defined as an isomorphic copy  ... 

Page 1028 of Mathematical Reviews Vol. 48, Issue 4 [page]

1974 Mathematical Reviews  
Two Steiner triple systems having the same set of points are said to be disjoint if they have no line in common.  ...  The interest in such a stems from the fact that the triples {a, b, «(b—a)+a} (a,b €S,a#b) form a Steiner triple system for which 8 acts as a regular, transitive automorphism group.  ... 

Page 2110 of Mathematical Reviews Vol. , Issue 97D [page]

1997 Mathematical Reviews  
A large set of Mendelsohn [directed] triple systems of order v is a collection of v —2 disjoint MTS(v) [DTS(v)] whose triples partition the set of all cyclic [directed] triples on the v-set.  ...  A 60 (1992), no. 2, 287-294; MR 93i:05024] that large sets of MTS and DTS exist for all v = 0,1 (mod 6) except for v = 6 in the case of a large set of MTS.  ... 

Page 3199 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
A large set of disjoint Kirkman triple systems of order v, or LKTS(wv), isa partition of the set of all 3-subsets of a v-set into v — 2 Kirkman triple systems of order v.  ...  For u = 1, the existence of a KS; (v; “4, 4) is equivalent to the existence of a doubly resolvable (v,k,2)-BIBD. The spectrum of KS;(v;1,1) or Room squares was completed by Mullin and Wallis in 1975.  ... 

Topological Effects in Tunneling-Coupled Systems of One-Dimensional Quantum Rings [article]

Colin Riggert, Kieran Mullen
2020 arXiv   pre-print
We find that writhe not only determines the energetic response of the system to magnetic field strength, but is also responsible for a single particle topological quantum phase transition involving the  ...  Using a model of idealized, crossed one-dimensional quantum wires we construct a novel model for a single electron on tunneling-coupled systems of one-dimensional quantum rings.  ...  (a) is the spectrum for the two ring system, and (b) is the spectrum for the figure eight system.  ... 
arXiv:2007.01967v1 fatcat:avvfbij2crbtrcty2hlnsyacre

Large sets of oriented triple systems with resolvability

Qing-de Kang, Zihong Tian
2000 Discrete Mathematics  
In this paper, we summarize the present situation about large sets (resp. overlarge sets) of several types of triple systems and give some new results.  ...  is a partition of X (or X \{x} for some x ∈ X ).  ...  The system is denoted by TKTS(v) = (X; B; G). The design TKTS(v) has played an important role to construct large sets of disjoint Kirkman triple systems.  ... 
doi:10.1016/s0012-365x(99)00288-5 fatcat:uurr5dhszzanjlvmhtme2xd3wm
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