Embeddings of pure mendelsohn triple systems and pure directed triple systems

It is proved in this article that the necessary and sufficient conditions for the embedding of a λ-fold pure Mendelsohn triple system of order v in λ-fold pure Mendelsohn triple of order u are λu(u ? 1) ≡ 0 (mod 3) and u ? 2v + 1. Similar results for the embeddings of pure directed triple systems are also obtained. © 1995 John Wiley & Sons, Inc.     

Large sets of resolvable MTS and DTS of orders pn + 2

An LRMTS(v) [resp., LRDTS(v)] is a large set consisting of v − 2 [resp., 3(v − 2)] disjoint resolvable Mendelsohn (resp., directed) triple systems of order v. In this article, we give a method to construct LRMTS(pⁿ + 2) and LRDTS(pⁿ + 2), where pⁿ is a prime power and pⁿ ≡ 1 (mod 6). Using the method and a recursive construction v → 3v, some unknown LRMTS(v) and LRDTS(v) are obtained such as for v = 69, 123, 141, 159, and 3^km, where k ≥ 1, m ϵ {7, 13, 37, 55, 57, 61, 65, 67}. © 1996 John Wiley & Sons, Inc.     

On the existence of nearly Kirkman triple systems with subsystems

It is proved in this paper that the necessary and sufficient conditions for the existence of an incomplete nearly Kirkman triple system INKTS(u, v) are u ≡ v ≡ 0 (mod 6), u ≥ 3v. As a consequence, we obtain a complete solution to the embedding problem for nearly Kirkman triple systems.      

Invariants and constructions of mendelsohn designs

In this paper we examine Mendelsohn designs and some connections to topology which lead to an easily described algorithm for computing invariants of these designs. The results are applied to designs which have natural group actions. We also use the topology to describe when ordinary two-fold triple systems with a group action lead to Mendelsohn designs with the same group action. Procedures for constructing Mendelsohn designs are also given. In particular, we give necessary and sufficient conditions for constructing 2-(v, 4, 1) Mendelsohn designs.     

More large sets of resolvable MTS and DTS with even orders

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.     

Doyen–Wilson theorem for perfect hexagon triple systems

The graph consisting of the three 3-cycles (or triples) (a,b,c), (c,d,e), and (e,f,a), where a,b,c,d,e and f are distinct is called a hexagon triple. The 3-cycle (a,c,e) is called an inside 3-cycle; and the 3-cycles (a,b,c), (c,d,e), and (e,f,a) are called outside 3-cycles. A hexagon triple system of order v is a pair (X,C), where C is a collection of edge disjoint hexagon triples which partitions the edge set of 3K_v. Note that the outside 3-cycles form a 3-fold triple system. If the hexagon triple system has the additional property that the collection of inside 3-cycles (a,c,e) is a Steiner triple system it is said to be perfect. In 2004, Küçükçifçi and Lindner had shown that there is a perfect hexagon triple system of order v if and only if and v≥7. In this paper, we investigate the existence of a perfect hexagon triple system with a given subsystem. We show that there exists a perfect hexagon triple system of order v with a perfect sub-hexagon triple system of order u if and only if v≥2u+1, and u≥7, which is a perfect hexagon triple system analogue of the Doyen–Wilson theorem.     

Existence of non‐resolvable Steiner triple systems

We consider two well‐known constructions for Steiner triple systems. The first construction is recursive and uses an STS(v) to produce a non‐resolvable STS(2v + 1), for v ≡ 1 (mod 6). The other construction is the Wilson construction that we specify to give a non‐resolvable STS(v), for v ≡ 3 (mod 6), v > 9. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 16–24, 2005.     

Quasi‐embeddings of Steiner triple systems,or Steiner triple systems of different orders with maximum intersection

In this paper, we present a conjecture that is a common generalization of the Doyen–Wilson Theorem and Lindner and Rosa's intersection theorem for Steiner triple systems. Given u, v ≡ 1,3 (mod 6), u < v < 2u + 1, we ask for the minimum r such that there exists a Steiner triple system such that some partial system can be completed to an STS , where |?| = r. In other words, in order to “quasi‐embed” an STS(u) into an STS(v), we must remove r blocks from the small system, and this r is the least such with this property. One can also view the quantity (u(u ? 1)/6) ? r as the maximum intersection of an STS(u) and an STS(v) with u < v. We conjecture that the necessary minimum r = (v ? u) (2u + 1 ? v)/6 can be achieved, except when u = 6t + 1 and v = 6t + 3, in which case it is r = 3t for t ≠ 2, or r = 7 when t = 2. Using small examples and recursion, we solve the cases v ? u = 2 and 4, asymptotically solve the cases v ? u = 6, 8, and 10, and further show for given v ? u > 2 that an asymptotic solution exists if solutions exist for a run of consecutive values of u (whose required length is no more than v ? u). Some results are obtained for v close to 2u + 1 as well. The cases where ≈ 3u/2 seem to be the hardest. © 2004 Wiley Periodicals, Inc.     

Maximal resolvable packings and minimal resolvable coverings of triples by quadruples

Determination of maximal resolvable packing number and minimal resolvable covering number is a fundamental problem in designs theory. In this article, we investigate the existence of maximal resolvable packings of triples by quadruples of order v (MRPQS(v)) and minimal resolvable coverings of triples by quadruples of order v (MRCQS(v)). We show that an MRPQS(v) (MRCQS(v)) with the number of blocks meeting the upper (lower) bound exists if and only if v≡0 (mod 4). As a byproduct, we also show that a uniformly resolvable Steiner system URS(3, {4, 6}, {r₄, r₆}, v) with r₆≤1 exists if and only if v≡0 (mod 4). All of these results are obtained by the approach of establishing a new existence result on RH(6²ⁿ) for all n≥2. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 209–223, 2010     

Small embeddings for partial 5‐cycle systems

It has been conjectured that any partial 5‐cycle system of order u can be embedded in a 5‐cycle system of order v whenever v ≥ 3 u/ 2+1 and v ≡ 1 , 5 (mod 10) . The smallest known embeddings for any partial 5‐cycle system of order u is 10 u +5 . In this paper we significantly improve this result by proving that for any partial 5‐cycle system of order u ≥ 255 , there exists a 5‐cycle system of order at most (9 u +146) / 4 into which the partial 5‐cycle system of order u can be embedded. © 2011 Wiley Periodicals, Inc. J Combin Designs     

On the embeddings of mendelsohn triple systems

It is proved in this paper that there exists an incomplete Mendelsohn triple system IMTS(u,v; λ) if and only ifλ(u-v)(u-2v-1)≡0(mod 3),u≥2v+1 and (u, v, λ) ≠ (6, 1, 1). As a consequence, it is proved that for any given λ≥1, a Mendelsohn triple system MTS (v, λ) can be embedded in an MTS (u, λ) if and only ifλu(u-1)≡0(mod 3) andu≥2v+1. Project supported by the National Natural Science Foundation of China.     

Tripling Construction for Large Sets of Resolvable Directed Triple Systems

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.     

Transitive resolvable idempotent quasigroups and large sets of resolvable Mendelsohn triple systems

We first define a transitive resolvable idempotent quasigroup (TRIQ), and show that a TRIQ of order v exists if and only if 3∣v and . Then we use TRIQ to present a tripling construction for large sets of resolvable Mendelsohn triple systems s, which improves an earlier version of tripling construction by Kang. As an application we obtain an for any integer n≥1, which provides an infinite family of even orders.     

Large sets of resolvable idempotent Latin squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the v(v−1) off-diagonal cells can be resolved into v−1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of v−2 RILS(v)s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.     

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, (mod 3), and . Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v – u – w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well‐known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. © 2005 Wiley Periodicals, Inc. J Combin Designs     

The existence of 2-balanced Mendelsohn triple systems

A Mendelsohn triple system MTS$(v,b)$ is a collection of $b$ cyclic triples (blocks) on a set of $v$ points. It is $j$-balanced for $j=1,2,3$ when any two points, ordered pairs, or cyclic triples (resp.) are contained in the same or almost the same number of blocks (difference at most one). A $(2,3)$-balanced Mendelsohn triple system is an MTS$(v,b)$ that is both 2-balanced and 3-balanced. Employing large sets of Mendelsohn triple systems and partitionable Mendelsohn candelabra systems, we completely determine the spectrum for which a 2-balanced Mendelsohn triple system exists. Meanwhile, we determine the existence spectrum for a $(2,3)$-balanced Mendelsohn triple system.     

Hybrid triple systems and cubic feedback sets

Ac-hybrid triple system of orderv is a decomposition of the completev-vertex digraph intoc cyclic tournaments of order 3 and transitive tournaments of order 3. Hybrid triple systems generalize directed triple systems (c = 0) and Mendelsohn triple systems (c = v(v – 1)/3); omitting directions yields an underlying twofold triple system. The spectrum ofv andc for which ac-hybrid triple system of orderv exists is completely determined in this paper. Using (cubic) block intersection graphs, we then show that every twofold triple system of order underlies ac-hybrid triple system with . Examples are constructed for all sufficiently largev, for which this maximum is at most . The lower bound here is proved by establishing bounds onF _i (n, r), the size of minimum cardinality vertex feedback sets inn-vertexi-connected cubic multigraphs havingr repeated edges. We establish that , 8$$ " align="middle" border="0"> . These bounds are all tight, and the latter is used to derive the lower bound in the design theoretic problem.     

RHPMDs and FHPMDs with k = 4 and Type 4 u

A (v, k, 1)-HPMD is called a frame (briefly, k-FHPMD), if the blocks of the HPMD can be partitioned into v partial parallel classes such that the complement of each partial parallel class is a group of the HPMD. A (v, k, 1)-HPMD is called resolvable (briefly, k-RHPMD), if the blocks of the HPMD can be partitioned into parallel classes. In this article, (i) we shall construct 3-FHPMDs of type 3⁶ and 21⁶ to completely settle the existence of 3-FHPMD of type h^u; (ii) we shall show that the necessary conditions for the existence of 4-FHPMD of type h^u are sufficient for the case h = 4; (iii) we shall show that the necessary conditions for the existence of 4-RHPMD of type h^u are sufficient for the case h = 4.     

A proof of Lindner's conjecture on embeddings of partial Steiner triple systems

Lindner's conjecture that any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if and is proved. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 63–89, 2009     

The existence of doubly near resolvable (v,3,2)-BIBDs

The existence of doubly near resolvable (v,2,1)-BIBDs was established by Mullin and Wallis in 1975. In this article, we determine the spectrum of a second class of doubly near resolvable balanced incomplete block designs. We prove the existence of DNR(v,3,2)-BIBDs for v ≡ 1 (mod 3), v ≥ 10 and v ? {34,70,85,88,115,124,133,142}. The main construction is a frame construction, and similar constructions can be used to prove the existence of doubly resolvable (v,3,2)-BIBDs and a class of Kirkman squares with block size 3, KS₃(v,2,4). © 1994 John Wiley & Sons, Inc.