Mendelsohn Triple System Research Articles

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.

Block‐avoiding point sequencings

AbstractLet and be positive integers. Recent papers by Kreher, Stinson, and Veitch have explored variants of the problem of ordering the points in a triple system (such as a Steiner triple system [STS], directed triple system, or Mendelsohn triple system) on points so that no block occurs in a segment of consecutive entries (thus the ordering is locally block‐avoiding). We describe a greedy algorithm which shows that such an ordering exists, provided that is sufficiently large when compared with . This algorithm leads to improved bounds on the number of points in cases where this was known, but also extends the results to a significantly more general setting (which includes, e.g., orderings that avoid the blocks of a design). Similar results for a cyclic variant of this situation are also established. We construct STSs and quadruple systems where can be large, showing that a bound of Stinson and Veitch is reasonable. Moreover, we generalize the Stinson–Veitch bound to a wider class of block designs and to the cyclic case. The results of Kreher, Stinson, and Veitch were originally inspired by results of Alspach, Kreher, and Pastine, who (motivated by zero‐sum avoiding sequences in abelian groups) were interested in orderings of points in a partial Steiner triple system where no segment is a union of disjoint blocks. Alspach et al. show that, when the system contains at most pairwise disjoint blocks, an ordering exists when the number of points is more than . By making use of a greedy approach, the paper improves this bound to .

Affine Mendelsohn triple systems and the Eisenstein integers

AbstractWe 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, Opršal, 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.

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.

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.

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.

Latin directed triple systems

It is well known that, given a Steiner triple system, a quasigroup can be formed by defining an operation ⋅ by the identities x ⋅ x = x and x ⋅ y = z , where z is the third point in the block containing the pair { x , y } . The same is true for a Mendelsohn triple system, where the pair ( x , y ) is considered to be ordered. But it is not true in general for directed triple systems. However, directed triple systems which form quasigroups under this operation do exist. We call these Latin directed triple systems, and in this paper we begin the study of their existence and properties.

The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups

AbstractA ternary quasigroup (or 3‐quasigroup) is a pair (N, q) where N is an n‐set and q(x, y, z) is a ternary operation on N with unique solvability. A 3‐quasigroup is called 2‐idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x)=y. A conjugation of a 3‐quasigroup, considered as an OA(3, 4, n), \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document}, is a permutation of the coordinate positions applied to the 4‐tuples of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\mathcal{B}}$\end{document}. The subgroup of conjugations under which \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document} is invariant is called the conjugate invariant subgroup of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document}. In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n, n≡7 or 11 (mod 12) and n≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n≡0, 1 (mod 3) and n≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010

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.

Frame self-orthogonal Mendelsohn triple systems of type [formula omitted

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 ( h n ) to denote an FSOMTS with n spanning holes of size h. The existence of FSOMTS ( h n ) for h ⩽ 3 has been known with a few exceptions. We extend the existing results and determine the necessary and sufficient conditions for the existence of FSOMTS ( h n ) for any h and n with some possible exceptions.

Frame Self-orthogonal Mendelsohn Triple Systems

A Mendelsohn triple system of order ν, MTS(ν) for short, is a pair (X, B) where X is a ν-set (of points) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B. The cyclic triple (a, b, c) contains the ordered pairs (a, b), (b, c) and (c, a). An MTS(ν) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order ν. An MTS(ν) is called frame self-orthogonal, FSOMTS for short, if its associated semisymmetric Latin square is frame self-orthogonal. It is known that an FSOMTS(1 n ) exists for all n≡1 (mod 3) except n=10 and for all n≥15, n≡0 (mod 3) with possible exception that n=18. In this paper, it is shown that (i) an FSOMTS(2 n ) exists if and only if n≡0,1 (mod 3) and n>5 with possible exceptions n∈{9, 27, 33, 39}; (ii) an FSOMTS(3 n ) exists if and only if n≥4, with possible exceptions that n∈{6, 14, 18, 19}.

Embedding extended Mendelsohn triple systems

In this paper, it is shown that every extended Mendelsohn triple system of order n and index λ can be embedded in an extended Mendelsohn triple system of order v> n and index λ if and only if v⩾2 n.

Large sets of disjoint pure Mendelsohn triple systems

A Mendelsohn triple system of order v , briefly MTS( v ), is a pair (X, B ) where X is a v -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( v ) is called pure and denoted by PMTS( v ) if 〈x,y,z〉∈ B implies 〈z,y,x〉∉ B . A large set of disjoint PMTS( v ) is denoted by LPMTS( v ). An overlarge set of PMTS( v ) is denoted by OLPMTS( v ). In this paper, some preliminary results about the existence and non-existence for LPMTS( v ) and OLPMTS( v ) are obtained.

Completing the spectrum of rotational Mendelsohn triple systems

A Mendelsohn triple system of order v is k-rotational if it has an automorphism consisting of a single fixed point and precisely k cycles of length ( v − 1)/ k. In this paper, we explore recursive constructions for rotational Mendelsohn triple systems which enable us to complete the existence problem of these systems.

The construction of antipodal triple systems by simulated annealing

An antipodal triple system of order v is a triple ( V, B, f), where | V | = v, B is a set of cyclically oriented 3-subsets of V, and f: V → V is an involution with one fixed point such that: 1. (i) ( V, B∪ f( B)) is a Mendelsohn triple system. 2. (ii) B ∩ f( B) = 0. 3. (iii) f is an isomorphism between the Steiner triple system ( STS) ( V, B′) and the STS ( V, f( B′)), where B′ is the same as B without orientation. 4. (iv) f preserves orientation. An STS (V,B) is hemispheric if there exists a cyclic orientation B ∗ of its block set B and an involution f such that (V,B ∗, f) is an antipodal system. We use simulated annealing on a carefully chosen feasibility space to show that any STS(v) (V,B), where 7 ⩽ v ⩽ 15, is hemispheric, and conjecture that this is true for any STS( v) v > 3. We were unable to find a way of applying the alternative computational techniques of hill climbing and backtracking to this problem.

Self-orthogonal mendelsohn triple systems

A Mendelsohn triple system of order v , briefly MTS( v ), is a pair ( X , B ) where X is a v -set (of points ) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B . The cyclic triple ( a, b, c ) contains the ordered pairs ( a, b ), ( b, c ), and ( c, a ). An MTS( v ) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order v . An MTS( v ) is called self-orthogonal , denoted briefly by SOMTS( v ), if its associated semisymmetric Latin square is self-orthogonal. It is well known that an MTS( v ) exists if and only if v ≡ 0 or 1 (mod 3) except v ≠ 6. It is also known that a SOMTS( v ) exists for all v ≡ 1 (mod 3) except v = 10 and that a SOMTS( v ) does not exist for v = 3, 6, 9, and 12. In this paper it is shown that a SOMTS( v ) exists for all v ⩾ 15, where v ≡ 0 (mod 3), except possibly for v = 18.

Self-Orthogonal Mendelsohn Triple Systems

A Mendelsohn triple system of order v , briefly MTS( v ), is a pair ( X , B ) where X is a v -set (of points ) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B . The cyclic triple ( a, b, c ) contains the ordered pairs ( a, b ), ( b, c ), and ( c, a ). An MTS( v ) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order v . An MTS( v ) is called self-orthogonal , denoted briefly by SOMTS( v ), if its associated semisymmetric Latin square is self-orthogonal. It is well known that an MTS( v ) exists if and only if v ≡ 0 or 1 (mod 3) except v ≠ 6. It is also known that a SOMTS( v ) exists for all v ≡ 1 (mod 3) except v = 10 and that a SOMTS( v ) does not exist for v = 3, 6, 9, and 12. In this paper it is shown that a SOMTS( v ) exists for all v ⩾ 15, where v ≡ 0 (mod 3), except possibly for v = 18.

Embeddings of pure mendelsohn triple systems and pure directed triple systems

AbstractIt 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.