Mendelsohn Triple Systems Research Articles

Decompositions of the λ-Fold Complete Mixed Graph into Mixed 6-Stars

Graph and digraph decompositions are a fundamental part of design theory. Probably the best known decompositions are related to decomposing the complete graph into 3-cycles (which correspond to Steiner triple systems), and decomposing the complete digraph into orientations of a 3-cycle (the two possible orientations of a 3-cycle correspond to directed triple systems and Mendelsohn triple systems). Decompositions of the λ-fold complete graph and the λ-fold complete digraph have been explored, giving generalizations of decompositions of complete simple graphs and digraphs. Decompositions of the complete mixed graph (which contains an edge and two distinct arcs between every two vertices) have also been explored in recent years. Since the complete mixed graph has twice as many arcs as edges, an isomorphic decomposition of a complete mixed graph into copies of a sub-mixed graph must involve a sub-mixed graph with twice as many arcs as edges. A partial orientation of a 6-star with two edges and four arcs is an example of such a mixed graph; there are five such mixed stars. In this paper, we give necessary and sufficient conditions for a decomposition of the λ-fold complete mixed graph into each of these five mixed stars for all λ>1.

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

An LPMTS(v,λ) is a collection of v−2λ disjoint pure Mendelsohn triple system PMTS(v,λ)s on the same set of v elements. An LPMTS⁎(v) is a special LPMTS(v,1) which contains exactly v−22 converse pairs of PMTS(v)s. In this paper, we mainly discuss the existence of an LPMTS⁎(v) for v≡6,10mod 12 and get the following conclusions: (1) there exists an LPMTS⁎(v) if and only if v≡0,4mod 6,v≥4 and v≠6. (2) There exists an LPMTS(v,λ) with index λ≡2,4mod 6 if and only if v≡0,4mod 6,v≥2λ+2,v≡2modλ.

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 gn exists when gn≡0(mod3) and (g,n)≠(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,ht) exists if and only if n≥3, t≥4 and nh(t−1)≡0(mod3), and then determine that the necessary condition for the existence of an RHMTS of type gn, namely, gn≡0(mod3) 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 existence of r-large sets of Mendelsohn triple systems

An r-large set of Mendelsohn triple systems of order v, denoted by r-LMTS(v), is a large set of ordered designs of order v (LOD(v)) which contains r MTS(v)s and v−r−23 cyclic tuples of OD(v)s. In this paper, some recursive constructions for r-LMTS(v)s are presented. Together with direct constructions, several infinite classes of r-LMTS(v)s are established.

Block-avoiding point sequencings of Mendelsohn triple systems

A cyclic ordering of the points in a Mendelsohn triple system of order v (or MTS(v)) is called a sequencing. A sequencing D is ℓ-good if there does not exist a triple (x,y,z) in the MTS(v) such that1. the three points x,y, and z occur (cyclically) in that order in D; and2. {x,y,z} is a subset of ℓ cyclically consecutive points of D.In this paper, we prove some upper bounds on ℓ for MTS(v) having ℓ-good sequencings and we prove that any MTS(v) with v≥7 has a 3-good sequencing. We also determine the optimal sequencings of every MTS(v) with v≤10.

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.

Cyclic and rotational latin hybrid triple systems

Abstract It is well known that given a Steiner triple system (STS) one can define a binary operation * upon its base set by assigning x * x = x for all x and x * y = z, where z is the third point in the block containing the pair {x, y}. The same can be done for Mendelsohn triple systems (MTS), directed triple systems (DTS) as well as hybrid triple systems (HTS), where (x, y) is considered to be ordered. In the case of STSs and MTSs the operation yields a quasigroup, however this is not necessarily the case for DTSs and HTSs. A DTS or an HTS which induces a quasigroup is said to be Latin. In this paper we study Latin DTSs and Latin HTSs which admit a cyclic or a 1-rotational automorphism. We prove the existence spectra for these systems as well as the existence spectra for their pure variants. As a side result we also obtain the existence spectra of pure cyclic and pure 1-rotational MTSs.

Distributive and Anti-distributive Mendelsohn Triple Systems

Abstract We prove that the existence spectrum of Mendelsohn triple systems whose associated quasigroups satisfy distributivity corresponds to the Loeschian numbers, and provide some enumeration results. We do this by considering a description of the quasigroups in terms of commutative Moufang loops. In addition we provide constructions of Mendelsohn quasigroups that fail distributivity for asmany combinations of elements as possible. These systems are analogues of Hall triple systems and anti-mitre Steiner triple systems respectively.

Antiflexible Latin directed triple systems

It is well known that given a Steiner triple system one can define a quasigroup operation $\cdot$ upon its base set by assigning $x \cdot x = x$ for all $x$ and $x \cdot y = z$, where $z$ is the third point in the block containing the pair $\{x,y\}$. The same can be done for Mendelsohn triple systems, where $(x,y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed triple systems. The quasigroups associated with Steiner and Mendelsohn triple systems satisfy the flexible law $y \cdot (x \cdot y) = (y \cdot x) \cdot y$ but those associated with Latin directed triple systems need not. In this paper we study the Latin directed triple systems where the flexible identity holds for the least possible number of ordered pairs $(x, y)$. We describe their geometry, present a surprisingly simple cyclic construction and prove that they exist if and only if the order $n$ is $n\equiv 0$ or $1\pmod{3}$ and $n\geq 13$.

A Completion of the Spectrum for the Overlarge Sets of Pure Mendelsohn Triple Systems

A pure Mendelsohn triple system of order v, denoted by PMTS(v), is a pair $$(X,\mathcal {B})$$(X,B) where X is a v-set and $$\mathcal {B}$$B is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of $$\mathcal {B}$$B and if $$\langle a,b,c\rangle \in \mathcal {B}$$?a,b,c??B implies $$\langle c,b,a\rangle \notin \mathcal {B}$$?c,b,a??B. An overlarge set of PMTS(v), denoted by OLPMTS(v), is a collection $$\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i$${(Y\{yi},Ai)}i, where Y is a $$(v+1)$$(v+1)-set, $$y_i\in Y$$yi?Y, each $$(Y{\setminus }\{y_i\},{\mathcal {A}}_i)$$(Y\{yi},Ai) is a PMTS(v) and these $${\mathcal {A}}_i$$Ais form a partition of all cyclic triples on Y. It is shown in [3] that there exists an OLPMTS(v) for $$v\equiv 1,3$$v?1,3 (mod 6), $$v>3$$v>3, or $$v \equiv 0,4$$v?0,4 (mod 12). In this paper, we shall discuss the existence problem of OLPMTS(v)s for $$v\equiv 6,10$$v?6,10 (mod 12) and get the following conclusion: there exists an OLPMTS(v) if and only if $$v\equiv 0,1$$v?0,1 (mod 3), $$v>3$$v>3 and $$v\ne 6$$v?6.

Large sets of resolvable Mendelsohn triple systems of prime power sizes

A large set of Mendelsohn triple systems of order v is a partition of all cyclic triples on a v-element set into pairwise disjoint Mendelsohn triple systems of order v. This note addresses questions related to the construction of large sets of Mendelsohn triple systems with resolvable property (each block set having a partition into parallel classes). An improved recursive method is established and a number of new infinite series large sets of prime power sizes are settled by recursion, in combination with already known small cases. To be specific, for all prime powers q<400 and q≡1(mod 3), large sets of resolvable Mendelsohn triple systems of order qn+2 are proved to exist for any positive integer n, possibly except for q=379,397.

New large sets of resolvable Mendelsohn triple systems

An LRMTS(v) is a large set consisting of v−2 pairwise disjoint resolvable Mendelsohn triple systems defined over the same v-element set. In this paper a new product construction for LRMTS is displayed by using generalized LR-designs. As well, several new existence families of LRMTS are constructed via 3-wise balanced designs. Finally the existence result of LRMTS is expanded by combining all the known constructions.

Triple systems and binary operations

It is well known that given a Steiner triple system (STS) one can define a binary operation ∗ upon its base set by assigning x∗x=x for all x and x∗y=z, where z is the third point in the block containing the pair {x,y}. The same can be done for Mendelsohn triple systems (MTSs) as well as hybrid triple systems (HTSs), where (x,y) is considered to be ordered. In the case of STSs and MTSs, the operation is a quasigroup, however this is not necessarily the case for HTSs. In this paper we study the binary operation induced by HTSs. It turns out that each such operation ∗ satisfies y∈{x∗(x∗y),(x∗y)∗x}andy∈{(y∗x)∗x,x∗(y∗x)} for all x and y from the base set. We call every binary operation that fulfils this condition hybridly symmetric.Not all idempotent hybridly symmetric operations can be obtained from HTSs. We show that these operations correspond to decompositions of a complete digraph into certain digraphs on three vertices. However, an idempotent hybridly symmetric quasigroup always comes from an HTS. The corresponding HTS is then called a latin HTS (LHTS). The core of this paper is the characterization of LHTSs and the description of their existence spectrum.

The intersection spectrum of hooked Skolem sequences and applications

A hooked Skolem sequence of order n is a sequence hSn=(s1,s2,…,s2n+1) of 2n+1 integers containing each of the integers 1,2,…,n exactly twice, such that two occurrences of the integer j∈{1,2,…,n} are separated by exactly j−1 integers, and s2n=0. We prove that the necessary conditions are sufficient for the existence of two hooked Skolem sequences of order n with 0,1,2,…,n−3 and n pairs in the same positions. Further, we apply this result to the fine structure of cyclic three-fold triple systems and cyclic four-fold triple systems for v≡13,19(mod24). Then, we extend these results to the fine structure of cyclic directed triple systems and cyclic Mendelsohn triple systems.

The Mendelsohn Triple Systems of Order 13

AbstractA triple cyclically contains the ordered pairs , , , and no others. A Mendelsohn triple system of order v, or , is a set V together with a collection of ordered triples of distinct elements from V, such that and each ordered pair with is cyclically contained in exactly λ ordered triples. By means of a computer search, we classify all Mendelsohn triple systems of order 13 with ; there are 6 855 400 653 equivalence classes of such systems.

Group divisible designs with block size four and group type [formula omitted] for more small [formula omitted

Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of {4}-GDDs of type gum1 is on the case when gu is even, where the existence for small g has played a key role. In order to determine the spectrum for {4}-GDDs of type gum1 with gu being odd, we continue to investigate the small cases with g∈{7,9,21} in this paper. We show that, for each g∈{7,9,21}, the necessary conditions for the existence of a {4}-GDD of type gum1 are also sufficient. As the applications of these GDDs, we obtain a few pairwise balanced designs with minimum block size 4. Meanwhile, we also improve the existence result for frame self-orthogonal Mendelsohn triple systems of type hn by reducing an infinite class of possible exceptions, namely n=9 and h≡2mod6, to eight undetermined cases.

Overlarge sets of Mendelsohn triple systems with resolvability

An OLRMTS(v) (OLARMTS(v)) over a (v+1)-set X is a collection {(X∖{x},Bx):x∈X} of v+1 pairwise disjoint resolvable (almost resolvable) Mendelsohn triple systems of order v. In this paper several direct construction methods for OLRMTSs and OLARMTSs are presented and then applied to produce some new orders; the smallest unknown OLRMTS(v) for v=18,24, the smallest unknown OLARMTS(v) for v=19,22,28,31, and some other small designs are displayed; a few new existence families are also obtained by known recursive constructions.

The intersection spectrum of Skolem sequences and its applications to [formula omitted]-fold cyclic triple systems

A Skolem sequence of order n is a sequence Sn=(s1,s2,…,s2n) of 2n integers containing each of the integers 1,2,…,n exactly twice, such that two occurrences of the integer j∈{1,2,…,n} are separated by exactly j−1 integers. We prove that the necessary conditions are sufficient for existence of two Skolem sequences of order n with 0,1,2,…,n−3 and n pairs in the same positions. Further, we apply this result to the fine structure of cyclic two-, three-, and four-fold triple systems, and also to the fine structure of λ-fold directed triple systems and λ-fold Mendelsohn triple systems.

Self-converse large sets of pure Mendelsohn triple systems

A Mendelsohn triple system of order υ (MTS(υ)) is a pair (X, ℬ) where X is a υ-set and ℬ is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of ℬ. An MTS(υ) (X, ℬ) is called pure and denoted by PMTS(υ) if 〈x, y, z〉 ∈ ℬ implies 〈z, y, x〉 ∉ ℬ. A large set of MTS(υ)s (LMTS(υ)) is a collection of υ − 2 pairwise disjoint MTS(υ)s on a υ-set. A self-converse large set of PMTS(υ)s, denoted by LPMTS*(υ), is an LMTS(υ) containing ⌎υ−2/2⌏ converse pairs of PMTS(υ)s. In this paper, some results about the existence and non-existence for LPMTS*(υ) are obtained.

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 v ⁄ ≡ 2 ( mod 4 ) . Then we use TRIQ to present a tripling construction for large sets of resolvable Mendelsohn triple systems LRMTS ( v ) s, which improves an earlier version of tripling construction by Kang. As an application we obtain an LRMTS ( 4 ⋅ 3 n ) for any integer n ≥ 1 , which provides an infinite family of even orders.