Constructions for cyclic 3‐designs and improved results on cyclic Steiner quadruple systems

This paper gives some recursive constructions for cyclic 3‐designs. Using these constructions we improve Grannell and Griggs's construction for cyclic Steiner quadruple systems, and many known recursive constructions for cyclic Steiner quadruple systems are unified. Finally, some new infinite families of cyclic Steiner quadruple systems are obtained. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:178‐201, 2011     

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 nelement 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).     

A lower bound on the number of Semi‐Boolean quadruple systems

A Steiner quadruple system of order 2ⁿ is Semi‐Boolean (SBQS(2ⁿ) in short) if all its derived triple systems are isomorphic to the point‐line design associated with the projective geometry PG(n?1, 2). We prove by means of explicit constructions that for any n, up to isomorphism, there exist at least 2^{? 3(n?4)/2?} regular and resolvable SBQS(2ⁿ). © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 229–239, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10050     

Spectrum of resolvable directed quadruple systems

A t-(v, k, 1) directed design (or simply a t-(v, k, 1)DD) is a pair (S, ℐ), where S is a v-set and ℐ is a collection of k-tuples (called blocks) of S, such that every t-tuple of S belongs to a unique block. The t-(v, k, 1)DD is called resolvable if ℐ can be partitioned into some parallel classes, so that each parallel class is a partition of S. It is proved that a resolvable 3-(v, 4, 1)DD exists if and only if v = 0 (mod 4).     

The Steiner quadruple systems of order 16

The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is 1,054,163. Properties of the designs—including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15—are tabulated. A consistency check based on double counting is carried out to gain confidence in the correctness of the classification.     

Resolvable candelabra quadruple systems with three groups

Candelabra quadruple systems play an important role in the construction of Steiner 3‐designs. In this article, we consider resolvable candelabra quadruple systems with three groups and show that the necessary conditions on the existence of RCQS(g³: s) when g is even are also sufficient. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:247‐267, 2011     

Pure tetrahedral quadruple systems with index two

An 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 $n$ with index $\lambda$, denoted by ${\text{TQS}}_{\lambda }\left(n\right)$, is a pair $\left(X,\mathrm{?}\right)$, where $X$ is an $n$‐set and $\mathrm{?}$ is a set of oriented tetrahedra (blocks) such that every cyclic triple on $X$ is contained in exactly $\lambda$ members of $\mathrm{?}$. A ${\text{TQS}}_{\lambda }\left(n\right)$ is pure if there do not exist two blocks with the same vertex set. When $\lambda =1$, the spectrum of a pure TQS $\left(n\right)$ has been completely determined by Ji. In this paper, we show that there exists a pure ${\text{TQS}}_2\left(n\right)$ if and only if $n\equiv 1,2\left(\mathrm{mod}3\right)$ and $n\ge 7$. A corollary is that a simple ${\text{QS}}_4\left(n\right)$ also exists if and only if $n\equiv 1,2\left(\mathrm{mod}3\right)$ and $n\ge 7$.     

Overlarge sets of disjoint Steiner quadruple systems

In this article, we construct overlarge sets of disjoint S(3, 4, 3ⁿ − 1) and overlarge sets of disjoint S(3, 4, 3ⁿ + 1) for all n ≥ 2. Up to now, the only known infinite sequence of overlarge sets of disjoint S(3, 4, v) were the overlarge sets of disjoint S(3, 4, 2ⁿ) obtained from the oval conics of desarguesian projective planes of order 2ⁿ. © 1999 John Wiley & Sons, Inc. J Combin Design 7: 311–315, 1999     

Existence of 3‐chromatic Steiner quadruple systems

A Steiner quadruple system of order v (briefly SQS (v)) is a pair (X, ), where X is a v‐element set and is a set of 4‐element subsets of X (called blocks or quadruples), such that each 3‐element subset of X is contained in a unique block of . The chromatic number of an SQS(v)(X, ) is the smallest m for which there is a map such that for all , where . The system (X, ) is equitably m‐chromatic if there is a proper coloring with minimal m for which the numbers differ from each other by at most 1. Linek and Mendelsohn showed that an equitably 3‐chromatic SQS(v) exists for v ≡ 4, 8, 10 (mod 12), v ≥ 16. In this article we show that an equitably 3‐chromatic SQS(v) exists for v ≡ 2 (mod 12) with v > 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 469–477, 2007     

On orthogonal resolutions of the classical Steiner quadruple system SQS(16)

A Steiner quadruple system SQS(16) is a pair where V is a 16-set of objects and is a collection of 4-subsets of V, called blocks, so that every 3-subset of V is contained in exactly one block. By classical is meant the boolean quadruple system, also known as the affine geometry AG(4,2). A parallel class is a collection of four blocks which partition V. The system possesses a resolution or parallelism, since can be partitioned into 35 parallel classes. Two resolutions are called orthogonal when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. We prove that there are at most nine further resolutions which, together with the classical one, are pairwise orthogonal.      

On the number of cyclic subgroups of a group

Abstract

There has been some interest in understanding the relationship between the number of cyclic subgroups of a group and its order. This relationship is controlled in many cases by the size of the set of non-squares of the group. We improve upon previously established bounds and classify groups that obtain our new bound.     

A note on conservative galaxies,Skolem systems,cyclic cycle decompositions,and Heffter arrays

The conservative number of a graph $G$ is the minimum positive integer $M$, such that $G$ admits an orientation and a labeling of its edges by distinct integers in $\{1,2,\dots ,M\}$, such that at each vertex of degree at least three, the sum of the labels on the in-coming edges is equal to the sum of the labels on the out-going edges. A graph is conservative if $M=\left|E\left(G\right)\right|$. It is worth noting that determining whether certain biregular graphs are conservative is equivalent to find integer Heffter arrays.In this work we show that the conservative number of a galaxy (a disjoint union of stars) of size $M$ is $M$ for $M\equiv 0$, $3(mod4)$, and $M+1$ otherwise. Consequently, given positive integers $m_1$, $m_2$, …, $m_n$ with $m_i\ge 3$ for $1\le i\le n$, we construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of infinitely many circulant graphs, generalizing a result of Bryant, Gavlas and Ling (2003). In particular, it allows us to construct a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of the complete graph $K_{2M+1}$, where $M={\sum }_{i=1}^n{m}_i$. Also, we prove necessary and sufficient conditions for the existence of a cyclic $(m_1,m_2,\dots ,m_n)$-cycle system of $K_{2M+2}?F$, where $F$ is a 1-factor. Furthermore, we give a sufficient condition for a subset of ${\mathbb{Z}}_v?\left\{0\right\}$ to be sequenceable.     

Jump relations of the quadruple layer potential on a regular surface in three dimensions

The jump relations of the quadruple layer potential on a regular surface in three dimensions are derived. The jumps are shown to be proportional to the product of the density of the potential and the mean curvature of the underlying surface.     

Directed cyclic Hamiltonian cycle systems of the complete symmetric digraph

In this paper, we prove that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph, , exist if and only if n is odd with n≠15 and n≠p^α for p an odd prime and α≥2 or with n≠2p^α for p an odd prime and α≥1. We also show that directed cyclic Hamiltonian cycle systems of the complete symmetric digraph minus a set of n/2 vertex-independent digons, (K_n−I)^∗, exist if and only if .     

The spectrum of the Burnside Tambara functor of a cyclic group

We derive a family of prime ideals of the Burnside Tambara functor for a finite group G. In the case of cyclic groups, this family comprises the entire prime spectrum. We include some partial results towards the same result for a larger class of groups.     

An improved product construction of rotational Steiner quadruple systems

An improved product construction is presented for rotational Steiner quadruple systems. Direct constructions are also provided for small orders. It is known that the existence of a rotational Steiner quadruple system of order υ+1 implies the existence of an optimal optical orthogonal code of length υ with weight four and index two. New infinite families of orders are also obtained for both rotational Steiner quadruple systems and optimal optical orthogonal codes. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 433–443, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10025