Mendelsohn Triple System Research Articles

Derived Mendelsohn triple systems

Mendelsohn triple system of order ν which can be extended to a tetrahedral quadruple system of order ν + 1 we call a derived Mendelsohn triple system. We consider some properties of derived Mendelsohn triple systems and give some results on their existence.

Cyclic antiautomorphisms of Mendelsohn triple systems

A cyclic triple ( a, b, c) is defined to be set {( a, b),( b, c),( c, a)} of ordered pairs. A Mendelsohn triple system of order v, M(2, 3, v), is a pair ( M, β), where M is a set of v points and β is a collection of cyclic triples of pairwise distinct points of M such that any ordered pair of distinct points of M is contained in precisely one cyclic triple of β. An antiautomorphism of a Mendelsohn triple system ( M, β) is a permutation of M which maps β to β -1, where β -1={( c, b, a); ( a, b, c) ϵβ}. In this paper we give necessary and sufficient conditions for the existence of a Mendelsohn triple system of order v admitting an antiautomorphism consisting of a single cycle of length d and having v - d fixed points. Further, we give a more general result for partial Mendelsohn triple systems in which the missing ordered pairs are precisely those containing two fixed points.

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.

Embeddings of resolvable mendelsohn triple systems

AbstractIn this article it is shown that any resolvable Mendelsohn triple system of order u can be embedded in a resolvable Mendelsohn triple system of order v iff v≥ 3u, except possibly for 71 values of (u,v). © 1993 John Wiley & Sons, Inc.

On indecomposable pure Mendelsohn triple systems

Let v and λ be positive integers. A Mendelsohn triple system MTS( v, λ) is a pair ( X, B ), where X is a v-set (of points) and B is a collection of cyclically ordered 3-subsets of X (called blocks or triples) such that every ordered pair of points of X is contained in exactly λ blocks of B . If we ignore the cyclic order of the blocks, then an MTS( v, λ) can be viewed as a ( v, 3, 2λ)-balanced incomplete block design (BIBD). An MTS( v, λ) is called pure if its underlying ( v, 3, 2λ)-BIBD contains no repeated blocks. An MTS( v, λ) is indecomposable if it is not the union of two Mendelsohn triple systems MTS( v, λ 1) and MTS( v, λ 2) with λ = λ 1 + λ 2. For λ = 2 and 3, the problem of existence of a pure MTS( v, λ) is completely solved in this paper. We further prove that an indecomposable pure MTS( v, 2) exists if and only if v = 0 or 1 (mod 3) and v ⩾ 6, except possibly v = 9, 12. We show that an indecomposable pure MTS( v, 3) exists for v = 8, 11, 14 and for all v ⩾ 17.

Embedding partial Mendelsohn triple systems

It is proved that a partial Mendelsohn Triple system on m symbols can be embedded in a Mendelsohn Triple System on t symbols for all admissible t⩾4 m, where if t is even then m⩾14.

On Mendelsohn and directed triple systems with repeated elements allowed in triples

The definitions of both Mendelsohn triple systems and directed (or transitive) triple systems are extended to include triples with repeated elements. It is shown that such systems exist if and only if the number of elements in the underlying set is divisible by three. Moreover, every balanced ternary design with suitable parameters may have its triples directed to form a directed (transitive) triple system with repeats. Finally provided the underlying set is of size at least nine, there exists a balanced ternary design on this set with block size three, the triples of which cannot be oriented to form a Mendelsohn triple system with repeats.