Hybrid Triple Systems Research Articles

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.

Self-converse large sets of pure hybrid triple systems

A hybrid triple system of order v, briefly by HTS (v), is a pair (X, B) where X is a v-set and B is a collection of cyclic and transitive triples (called blocks) on X such that every ordered pair of X belongs to exactly one block of B. An HTS (v) is called pure and denoted by PHTS (v) if one element of the block set {(x, y, z), (z, y, x), (z, x, y), (y, x, z), (y, z, x), (x, z, y), 〈x, y, z〉, 〈z, y, x〉} is contained in B then the others will not be contained in B. A self-converse large set of disjoint PHTS (v)s, denoted by LPHTS*(v), is a collection of 4(v − 2) disjoint PHTS (v)s which contains exactly (v − 2)/2 converse octads of PHTS (v)s. In this paper, some results about the existence for LPHTS*(v) are obtained.

The existence spectrum for large sets of pure Hybrid triple systems

An LPHTS(v) is a collection of 4(v−2) disjoint pure Hybrid triple systems on the same set of v elements. In Fan (2010), it is showed that there exists an LPHTS(v) for v≡0,4 mod 6. In this paper, we establish the existence of an LPHTS(v) for v≡1,3 mod 6, v>3. Finally, the spectrum for LPHTSs is completely determined.

The Existence Spectrum for Overlarge Sets of Pure Hybrid Triple Systems

An overlarge set of pure Hybrid triple system $$(PHTS)$$(PHTS), denoted by $$OLPHTS(v)$$OLPHTS(v), is a collection $$\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i$${(Y\{yi},Ai)}i, where $$Y$$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 $$PHTS(v)$$PHTS(v) and these $${\mathcal {A}}_i$$Ais form a partition of all cyclic triples and transitive triples on $$Y.$$Y. In this paper, we shall discuss the existence problem of $$OLPHTSs$$OLPHTSs and get the following conclusion: there exists an $$OLPHTS(v)$$OLPHTS(v) if and only if $$v\equiv 0,1$$v?0,1 mod 3 and $$v>3$$v>3.

The spectrum for overlarge sets of hybrid triple systems

A hybrid triple system of order v and index λ, denoted by HTS(v, λ), is a pair (X, B) where X is a v-set and B is a collection of cyclic triples and transitive triples on X, such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint HTS(v, λ), denoted by OLHTS(v, λ), is a collection {(Y\ {y}, Ai)}i, such that Y is a (v + 1)-set, each (Y \{y}, Ai) is an HTS(v, λ) and all Ais form a partition of all cyclic triples and transitive triples on Y. In this paper, we shall discuss the existence problem of OLHTS(v, λ) and give the following conclusion: there exists an OLHTS(v, λ) if and only if λ = 1, 2, 4, v ≡ 0,1 (mod 3) and v ≥ 4.

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.

Cyclic and rotational hybrid triple systems

A hybrid triple system of order ν, denoted HTS(ν), is said to be cyclic if it admits an automorphism consisting of a single cycle of length ν. A HTS(ν) admitting an automorphism consisting of a fixed point and a cycle of length ν − 1 is said to be rotational. Necessary and sufficient conditions are given for the existence of a cyclic HTS(ν) and a rotational HTS(ν).

Cyclic and rotational hybrid triple systems

A hybrid triple system of order v, denoted HTS( v), is said to be cyclic if it admits an automorphism consisting of a single cycle of length v. A HTS( v) admitting an automorphism consisting of a fixed point and a cycle of length v − 1 is said to be rotational. Necessary and sufficient conditions are given for the existence of a cyclic HTS( v) and a rotational HTS( v).

Large sets of three types of oriented triple systems

Mendelsohn triple systems, directed triple systems and hybrid triple systems are three types of oriented triple systems. The existence spectrums for the large sets of these triple systems have been completed successively. This paper summarizes the process to complete these spectrums.

On large sets of hybrid triple systems

An oriented triple system B on a ν-set X is called an HTS(ν,λ) if it contains both cyclic and transitive triples such that each ordered pair of distinct elements is contained in exactly λ triples of B . If all the cyclic and transitive triples from X can be partitioned into ∪ r B r such that each (X, B r is an HTS( ν, λ), then {(X, B r} r is called an LHTS(ν,λ). In this paper, the existence spectrum of LHTS(ν,λ) is completed, that is 4( ν − 2) ≡ 0 (mod λ), 4( ν − 2) ⩾ λ, if λ ≉ 0 (mod 3) then ν ≉ 2 (mod 3) and if λ = 1 then ν ≠ 3.

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 $$\frac{{v(v - 1)}}{3} - c$$ 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 $$v\left( {having b_v = \frac{{v(v - 1)}}{3}blocks} \right)$$ underlies ac-hybrid triple system with $$c \geqslant \frac{{2b_v }}{3}$$ . Examples are constructed for all sufficiently largev, for which this maximum is at most $$\left( {\frac{7}{{10}} + \varepsilon } \right)b_v $$ . 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 $$F_0 (n,r) \leqslant \frac{n}{2},$$ , $$F_1 (n,r) \leqslant \frac{{3n}}{8} + \frac{r}{4} + \frac{1}{2}, and F_2 (n,r) \leqslant \frac{{(n + r)}}{3}for n > 8$$ . These bounds are all tight, and the latter is used to derive the lower bound in the design theoretic problem.