Resolvable Mendelsohn designs with block size 4

Summary Letv andK be positive integers. A (v, k, 1)-Mendelsohn design (briefly (v, k, 1)-MD) is a pair (X,B) whereX is av-set (ofpoints) andB is a collection of cyclically orderedk-subsets ofX (calledblocks) such that every ordered pair of points ofX are consecutive in exactly one block ofB. A necessary condition for the existence of a (v, k, 1)-MD isv(v–1) 0 (modk). If the blocks of a (v, k, 1)-MD can be partitioned into parallel classes each containingv/k blocks wherev ) (modk) or (v – 1)/k blocks wherev 1 (modk), then the design is calledresolvable and denoted briefly by (v, k, 1)-RMD. It is known that a (v, 3,1)-RMD exists if and only ifv 0 or 1 (mod 3) andv 6. In this paper, it is shown that the necessary condition for the existence of a (v, 4, 1)-RMD, namelyv 0 or 1 (mod 4), is also sufficient, except forv = 4 and possibly exceptingv = 12. These constructions are equivalent to a resolvable decomposition of the complete symmetric directed graphK _v ^* onv vertices into 4-circuits.Research supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-5320.     

Frame self-orthogonal Mendelsohn triple systems of type

A Mendelsohn triple system (MTS) corresponds to an idempotent semisymmetric Latin square (quasigroup) of the same order. A holey MTS is called frame self-orthogonal, briefly FSOMTS, if its associated holey semisymmetric Latin square is frame self-orthogonal. In this paper, we use FSOMTS(hⁿ) to denote an FSOMTS with n spanning holes of size h. The existence of FSOMTS(hⁿ) for h3 has been known with a few exceptions. We extend the existing results and determine the necessary and sufficient conditions for the existence of FSOMTS(hⁿ) for any h and n with some possible exceptions.     

Self-converse large sets of pure Mendelsohn triple systems

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.     

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.     

Affine Mendelsohn triple systems and the Eisenstein integers

We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to be affine, nonramified. By exhibiting a one‐to‐one correspondence between isomorphism classes of affine MTS and those of modules over the Eisenstein integers, we solve the isomorphism problem for affine, nonramified MTS and enumerate these isomorphism classes (extending the work of Donovan, Griggs, McCourt, Opr?al, and Stanovský). As a consequence, all entropic MTSs of order coprime with 3 and distributive MTS of order coprime with 3 are classified. Partial results on the isomorphism problem for affine MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any affine MTS, the qualities of being nonramified, pure, and self‐orthogonal are equivalent.     

The spectrum of resolvable holey Mendelsohn triple systems and holey Mendelsohn frames with block size three

A holey Mendelsohn triple system (HMTS) is a decomposition of a complete multipartite directed graph into directed cycles of length 3. If the directed cycles of length 3 can be partitioned into parallel classes, then the HMTS is called an RHMTS. Bennett, Wei and Zhu [J. Combin. Des., 1997] showed that an RHMTS of type $g^n$ exists when $gn\equiv 0(\mathrm{mod}3)$ and $(g,n)\ne (1,6)$ with some possible exceptions. In this paper, motivated by the application in constructing RHMTSs, we investigate the constructions of holey Mendelsohn frames. We prove that a 3-MHF of type $(n,h^t)$ exists if and only if $n\ge 3$, $t\ge 4$ and $nh(t-1)\equiv 0(\mathrm{mod}3)$, and then determine that the necessary condition for the existence of an RHMTS of type $g^n$, namely, $gn\equiv 0(\mathrm{mod}3)$ is also sufficient except for $(g,n)=(1,6)$. New recursive constructions on incomplete RHMTSs via MHFs are introduced to settle this problem completely.     

The spectrum for large sets of pure Mendelsohn triple systems

An LPMTS(v) is a collection of v-2 disjoint pure Mendelsohn triple systems on the same set of v elements. In this paper, the concept of t-purely partitionable Mendelsohn candelabra system (or t-PPMCS in short) is introduced for constructing LPMTS(v)s. A powerful recursive construction for t-PPMCSs is also displayed by utilizing s-fan designs. Together with direct constructions, the existence of an LPMTS(v) for and v>1 is established. For odd integer v?7, a special construction from both LPMTS(v) and OLPMTS(v) to LPMTS(2v+1) is set up. Finally, the existence of an LPMTS(v) is completely determined to be the set .     

Further results on large set of disjoint pure Mendelsohn triple systems

In this article, we study a large set of disjoint pure Mendelsohn triple systems “with holes” (briefly LPHMTS), which is a generalization of large set of disjoint pure Mendelsohn triple systems (briefly LPMTS), and give some recursive constructions on LPHMTS. Using these constructions, we show that there exists LPMTS(2ⁿ + 2) for any n ≠ 2. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 274–290, 2000     

The spectrum for self-converse large set of pure Mendelsohn triple systems

An $LPMTS(v,\lambda )$ is a collection of $\frac{v-2}{\lambda }$ disjoint pure Mendelsohn triple system $PMTS(v,\lambda )s$ on the same set of v elements. An $LPMT{S}^{⁎}(v)$ is a special $LPMTS(v,1)$ which contains exactly $\frac{v-2}{2}$ converse pairs of $PMTS(v)s$. In this paper, we mainly discuss the existence of an $LPMT{S}^{⁎}(v)$ for $v\equiv 6,10\mathrm{mod 12}$ and get the following conclusions: (1) there exists an $LPMT{S}^{⁎}(v)$ if and only if $v\equiv 0,4\mathrm{mod 6},v\ge 4$ and $v\ne 6$. (2) There exists an $LPMTS(v,\lambda )$ with index $\lambda \equiv 2,4\mathrm{mod 6}$ if and only if $v\equiv 0,4\mathrm{mod 6},v\ge 2\lambda +2,v\equiv 2\mathrm{mod}\mathit{\lambda }$.     

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.     

Perfect Mendelsohn designs with equal-sized holes and block size four

Let M = {m₁, m₂, …, m_h} and X be a v-set (of points). A holey perfect Mendelsohn designs (briefly (v, k, λ) - HPMD), is a triple (X, H, B), where H is a collection of subsets of X (called holes) with sizes M and which partition X, and B is a collection of cyclic k-tuples of X (called blocks) such that no block meets a hole in more than one point and every ordered pair of points not contained in a hole appears t-apart in exactly λ blocks, for 1 ≤ t ≤ k − 1. The vector (m₁, m₂, …, m_h) is called the type of the HPMD. If m₁ = m₂ = … = m_h = m, we write briefly m^h for the type. In this article, it is shown that the necessary condition for the existence of a (v, 4, λ) - HPMD of type m^h, namely, is also sufficient with the exception of types 2⁴ and 1⁸ with λ = 1, and type m⁴ for odd m with odd λ. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 203–213, 1997     

Resolvable candelabra quadruple systems with three groups

Candelabra quadruple systems play an important role in the construction of Steiner 3‐designs. In this article, we consider resolvable candelabra quadruple systems with three groups and show that the necessary conditions on the existence of RCQS(g³: s) when g is even are also sufficient. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:247‐267, 2011     

The Existence of (ν,6, λ)-Perfect Mendelsohn Designs with λ > 1

The basic necessary conditions for the existence of a (v, k, λ)-perfect Mendelsohn design (briefly (v, k, λ)-PMD) are v ≥ k and λ v(v − 1) ≡ 0 (mod k). These conditions are known to be sufficient in most cases, but certainly not in all. For k = 3, 4, 5, 7, very extensive investigations of (v, k, λ)-PMDs have resulted in some fairly conclusive results. However, for k = 6 the results have been far from conclusive, especially for the case of λ = 1, which was given some attention in papers by Miao and Zhu [34], and subsequently by Abel et al. [1]. Here we investigate the situation for k = 6 and λ > 1. We find that the necessary conditions, namely v ≥ 6 and λ v(v − 1)≡0 (mod 6) are sufficient except for the known impossible cases v = 6 and either λ = 2 or λ odd. Researcher F.E. Bennett supported by NSERC Grant OGP 0005320.     

Embedding Steiner triple systems in hexagon triple systems

A hexagon triple is the graph consisting of the three triangles (triples) {a,b,c},{c,d,e}, and {e,f,a}, where a,b,c,d,e, and f are distinct. The triple {a,c,e} is called an inside triple. A hexagon triple system of order n is a pair (X,H) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K_n with vertex set X. The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3n.     

Transverse quadruple systems with five holes

Transverse Steiner quadruple systems with five holes are either of type g⁵ or g⁴u¹. We concentrate on the systems of type g⁴u¹ and settle existence except when g ≡ u ≡ 2 (mod 4) and all except 40 parameter situations when g ≡ u + 2 ≡ 0 (mod 4). The question of existence for transverse quadruple systems of type g⁴u¹ with index λ > 1 is completely solved for all λ ≥ 13 and λ ∈ {4, 6, 8, 9, 10, 12}. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 315–340, 2007