Cyclic Triples Research Articles

Pure tetrahedral quadruple systems with index two

AbstractAn oriented tetrahedron defined on four vertices is a set of four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order with index , denoted by , is a pair , where is an ‐set and is a set of oriented tetrahedra (blocks) such that every cyclic triple on is contained in exactly members of . A is pure if there do not exist two blocks with the same vertex set. When , the spectrum of a pure TQS has been completely determined by Ji. In this paper, we show that there exists a pure if and only if and . A corollary is that a simple also exists if and only if and .

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.

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.

On the Number of 4-Cycles in a Tournament

AbstractIf T is an n-vertex tournament with a given number of 3-cycles, what can be saidabout the number of its 4-cycles? The most interesting range of this problem is whereT is assumed to have c⋅ n 3 cyclic triples for some c> 0 and we seek to minimize thenumber of 4-cycles. We conjecture that the (asymptotic) minimizing T is a randomblow-up of a constant-sized transitive tournament. Using the method of flag algebras,we derive a lower bound that almost matches the conjectured value. We are able toanswer the easier problem of maximizing the number of 4-cycles. These questions canbe equivalently stated in terms of transitive subtournaments. Namely, given the numberof transitive triples in T, how many transitive quadruples can it have? As far as weknow, this is the first study of inducibility in tournaments. 1 Introductionand notation 1.1 Notation For tournaments T,H, let pr(H,T) be the probability that a random set of SHS vertices inT spans a subtournament isomorphic to H. For an infinite family of tournaments T , letpr(H,T ) = lim

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.

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.

What shall we do with the cyclic profile?

A Condorcet method for a single-seat preferential election is defined by the way it handles cases without a Condorcet winner. Such cases are described in terms of cyclic candidate triples. Sometimes election theory considers intricate patterns of intersecting cyclic triples. In practice, cyclic triples are rare, but they may be created as part of strategic voting against a Condorcet winner. This strategy, often called burying, is difficult to apply, but critics will detect a missed opportunity and regard it as a punishment for honest voting. A new Condorcet method, SV, lets a cyclic triple be won by the candidate with the smallest opportunity to defeat a Condorcet winner this way. A new geometric model allows visual comparison of the opportunities under various methods.

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.

Purely tetrahedral quadruple systems

An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X, B), where X is an n-element set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X, B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n ≡ 2,4 (mod 6), n > 4, or n ≡ 1, 5 (mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n ≡ 2,4 (mod 6) and n > 4, or n ≡ 1,5 (mod 12). Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n ≡ 1,3 (mod 6), n > 3, or n ≡ 0,4 (mod 12).

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}.

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.

On the Maximum Number of Cyclic Triples in Oriented Graphs

The maximum number of cyclic triples in an oriented graph of a given order is well known, being realized by the regular and near-regular tournaments. We consider the corresponding problem of determining the maximum number of cyclic triples in an oriented graph with a given number of arcs. A formula is found provided that the number of vertices is not too small relative to the number of arcs.

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.

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.

Simple direct constructions for hybrid triple designs

A c-hybrid triple design, c-HTD( v, λ), is a partition of the arcs of the complete symmetric digraph in which each arc has multiplicity λ, λDK v , into c cyclic triples and λv(v − 1)/3 − c transitive triples. We give a simple direct construction for all such designs.

A computer program to evaluate paired-comparison transitivity

The method of paired comparisons is a useful assessment technique when a simple task is required or when the underlying transitivity of elements being compared is suspect. The transitivity of elements can be assessed by evaluating all possible triples or, where n signifies the number of elements being compared, n(n -1)(n 2)16 triples. The number of transitive triples (e.g., A> B, B > C, A> C) can be determined from the row marginals of the pairedcomparison scoring matrix (see Harary, Norman, & Cartwright, 1965), but determining exactly which triples are transitive or cyclic (A > B, B > C, C > A) can be a formidable task. When n is 5, the number of triples is a manageable 10, but when n is 10, the number of triples is 120, and when n is 20, the number of triples is 1,140. The present program, written in FORTRAN IV, evaluates all possible triples, prints all triples that are cyclic, and provides several statistics relating to transitivity. Input. The first card indicates the size (i.e., order) of the paired-comparison scoring matrix (n can range from 3 to 25), followed by cards representing the paired-comparison scoring matrix; each card signifies one row of the matrix. In a paired-comparison scoring matrix, an entry of 1 in the ith row and jth column indicates that the ith element was preferred to the jth element; an entry of 0 indicates the opposite. Output. The output consists of the printing of input data, followed by a listing of all cyclic triples and indexes relating to transitivity, The following is an example of a cyclic triple:

Extended cyclic triple systems

An extended cyclic triple system is a pair ( S, W) where S is a finite set and W is a collection of cyclic triples from S, where each triple may have repeated elements, such that every ordered pair of elements of S, not necessarily distinct, is contained in exactly one triple on W. The triples of W are of three types: (1) { a, a, a}, (2) { b, b, c}, (3) { x, y, z}, where { b, b, c} contains the pairs bc, bc and cb, while { x, y, z} contains xy, yz and zx but not xz, yx or zy. The element z is called an idempotent and b a non-idempotent of ( S, W). We denote by CTS ( n; α) be class of all extended cyclic triple systems on n elements which have α idempotents. We say CTS ( n; α) exists if there is a system with parameters n and α. We investigate conditions for the existence of CTS ( n; α) and show that if the necessary conditions are satisfied then CTS( n; α) exists.

Generating the alternating group by cyclic triples

It is shown that a collection of circular permutations of length three on an n-set generates the alternating group A n if and only if the associated graph is connected. It follows that [ 1 2 n ] circular permutations of length three may generate A n .