Steiner Triple Systems Research Articles

Nestings of BIBDs with block size four

AbstractIn a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a ‐BIBD in such a way that we end up with a partial ‐BIBD. In the case where the partial ‐BIBD is in fact a ‐BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if . Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., ‐BIBDs) when , as well as for some symmetric BIBDs. Here we study nestings of ‐BIBDs, which are not perfect nestings. We prove that there is a nested ‐BIBD if and only if , . This is accomplished by a variety of direct and recursive constructions.

Cancellative hypergraphs and Steiner triple systems

A triple system is cancellative if it does not contain three distinct sets A,B,C such that the symmetric difference of A and B is contained in C. We show that every cancellative triple system H that satisfies a particular inequality between the sizes of H and its shadow must be structurally close to the balanced blowup of some Steiner triple system. Our result contains a stability theorem for cancellative triple systems due to Keevash and Mubayi as a special case. It also implies that the boundary of the feasible region of cancellative triple systems has infinitely many local maxima, thus giving the first example showing this phenomenon.

Large monochromatic components in expansive hypergraphs

AbstractA result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary $r$ -colouring of the complete $k$ -uniform hypergraph $K_n^k$ when $k\geq 2$ and $k\in \{r-1,r\}$ . We prove a result which says that if one replaces $K_n^k$ in Gyárfás’ theorem by any ‘expansive’ $k$ -uniform hypergraph on $n$ vertices (that is, a $k$ -uniform hypergraph $G$ on $n$ vertices in which $e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets $V_1, \ldots, V_k\subseteq V(G)$ with $|V_i|\gt \alpha$ for all $i\in [k]$ ), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on $r$ and $\alpha$ ). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary $r$ -partite $r$ -uniform hypergraph $H$ with $n$ edges in which every set of $k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of $k$ edges has a common intersection with the condition that for every collection of $k$ disjoint sets $E_1, \ldots, E_k\subseteq E(H)$ with $|E_i|\gt \alpha$ , there exists $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that $e_1\cap \cdots \cap e_k\neq \emptyset$ , then the smallest possible maximum degree of $H$ is essentially the same (within a small error term depending on $r$ and $\alpha$ ). We prove our results in this dual setting.

Decompositions of the λ-Fold Complete Mixed Graph into Mixed 6-Stars

Graph and digraph decompositions are a fundamental part of design theory. Probably the best known decompositions are related to decomposing the complete graph into 3-cycles (which correspond to Steiner triple systems), and decomposing the complete digraph into orientations of a 3-cycle (the two possible orientations of a 3-cycle correspond to directed triple systems and Mendelsohn triple systems). Decompositions of the λ-fold complete graph and the λ-fold complete digraph have been explored, giving generalizations of decompositions of complete simple graphs and digraphs. Decompositions of the complete mixed graph (which contains an edge and two distinct arcs between every two vertices) have also been explored in recent years. Since the complete mixed graph has twice as many arcs as edges, an isomorphic decomposition of a complete mixed graph into copies of a sub-mixed graph must involve a sub-mixed graph with twice as many arcs as edges. A partial orientation of a 6-star with two edges and four arcs is an example of such a mixed graph; there are five such mixed stars. In this paper, we give necessary and sufficient conditions for a decomposition of the λ-fold complete mixed graph into each of these five mixed stars for all λ>1.

Dominator coloring of Kneser graph

A dominator coloring (briefly DC) of a graph G is a proper coloring such that each vertex dominates every vertex of at least one color class (possibly its own class). The dominator chromatic number χd(G) of G is the smallest cardinality of color classes in a DC of G. By using Steiner triple systems, in this paper, we establish that the dominator coloring of the Kneser graph KG(n, 2) is n for n ≥ 5.

Steiner triple systems of order 21 with subsystems

The smallest open case for classifying Steiner triple systems is order 21. A Steiner triple system of order 21, an STS\((21)\), can have subsystems of orders 7 and 9, and it is known that there are 12,661,527,336 isomorphism classes of STS\((21)\)s with sub-STS\((9)\)s. Here, the classification of STS\((21)\)s with subsystems is completed by settling the case of STS\((21)\)s with sub-STS\((7)\)s. There are 116,635,963,205,551 isomorphism classes of such systems. An estimation of the number of isomorphism classes of STS\((21)\)s is given.

New bounds on the size of nearly perfect matchings in almost regular hypergraphs

AbstractLet be a ‐uniform ‐regular simple hypergraph on vertices. Based on an analysis of the Rödl nibble, in 1997, Alon, Kim and Spencer proved that if , then contains a matching covering all but at most vertices, and asked whether this bound is tight. In this paper we improve their bound by showing that for all , contains a matching covering all but at most vertices for some , when and are sufficiently large. Our approach consists of showing that the Rödl nibble process not only constructs a large matching but it also produces many well‐distributed ‘augmenting stars’ which can then be used to significantly improve the matching constructed by the Rödl nibble process. Based on this, we also improve the results of Kostochka and Rödl from 1998 and Vu from 2000 on the size of matchings in almost regular hypergraphs with small codegree. As a consequence, we improve the best known bounds on the size of large matchings in combinatorial designs with general parameters. Finally, we improve the bounds of Molloy and Reed from 2000 on the chromatic index of hypergraphs with small codegree (which can be applied to improve the best known bounds on the chromatic index of Steiner triple systems and more general designs).

The first infinite family of orthogonal Steiner systems S(3,5,v)

The research on orthogonal Steiner systems S(t,k,v) was initiated in 1968. For (t,k)∈{(2,3),(3,4)}, this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t≥3 and k≥5, only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S(3,5,v) is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems S(3,5,4m+1) is established for any even m≥2; additionally a pair of orthogonal G-designs G(4m+15,5,5,3) is displayed for any odd m≥3. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S(5,8,24) are shown to exist.

Block-Graceful Designs

In this article, we adapt the edge-graceful graph labeling definition into block designs and define a block design V , B with V = v and B = b as block-graceful if there exists a bijection f : B ⟶ 1,2 , … , b such that the induced mapping f + : V ⟶ Z v given by f + x = ∑ x ∈ A A ∈ B f A mod v is a bijection. A quick observation shows that every v , b , r , k , λ − BIBD that is generated from a cyclic difference family is block-graceful when v , r = 1 . As immediate consequences of this observation, we can obtain block-graceful Steiner triple system of order v for all v ≡ 1 mod 6 and block-graceful projective geometries, i.e., q d + 1 − 1 / q − 1 , q d − 1 / q − 1 , q d − 1 − 1 / q − 1 − BIBDs. In the article, we give a necessary condition and prove some basic results on the existence of block-graceful v , k , λ − BIBDs. We consider the case v ≡ 3 mod 6 for Steiner triple systems and give a recursive construction for obtaining block-graceful triple systems from those of smaller order which allows us to get infinite families of block-graceful Steiner triple systems of order v for v ≡ 3 mod 6 . We also consider affine geometries and prove that for every integer d ≥ 2 and q ≥ 3 , where q is an odd prime power or q = 4 , there exists a block-graceful q d , q , 1 − BIBD. We make a list of small parameters such that the existence problem of block-graceful labelings is completely solved for all pairwise nonisomorphic BIBDs with these parameters. We complete the article with some open problems and conjectures.

A visual representation of the Steiner triple systems of order 13

Steiner triple systems (STSs) are a basic topic in combinatorics. In an STS the elements can be collected in threes in such a way that any pair of elements is contained in a unique triple. The two smallest nontrivial STSs, with 7 and 9 elements, arise in the context of finite geometry and nonsingular cubic curves, and have well-known pictorial representations. On the contrary, an STS with 13 elements does not have an intrinsic geometric nature, nor a natural pictorial illustration. In this paper we present a visual representation of the two non-isomorphic Steiner triple systems of order 13 by means of a regular hexagram. The thirteen points of each system are the vertices of the twelve equilateral triangles inscribed in the hexagram. In the case of the non-cyclic system, our representation allows one to visualize in a simple, elegant and highly symmetric way the twenty-six triples, the six automorphisms and their orbits, the eight quadrilaterals, the ten mitres, the thirteen grids, the four 3-colouring patterns, the block-colouring and some distinguished ovals. Our construction is based on a very simple idea (seeing the blocks as much as possible as equilateral triangles), which can be further extended to get new representations of the STSs of order 7 and 9, and of one of the STSs of order 15.

The spectrum of resolvable Bose triple systems

A classical construction of Bose produces a Steiner triple system of order 3n from a symmetric, idempotent latin square of order n, which exists whenever n is odd. In an application to access-balancing in storage systems, certain Bose triple systems play a central role. A natural question arises: For which orders v does there exist a resolvable Bose triple system? Elementary counting establishes the necessary condition that v≡9(mod18). For specific Bose triple systems that optimize an access metric, we show that v≡9(mod18) is also sufficient.

On the seven non-isomorphic solutions of the fifteen schoolgirl problem

In this paper we give a simple and effective tool to analyze a given Kirkman triple system of order 15 and determine which of the seven well-known non-isomorphic KTS(15)s it is isomorphic to.Our technique refines and improves the lacing of distinct parallel classes introduced by F. N. Cole, by means of the notion of residual triple defined by G. Falcone and the present author in a previous paper.Unlike Cole's original lacing scheme, our algorithm allows one to distinguish two KTS(15)s also in the harder case where the two systems have the same underlying Steiner triple system. In the special case where the common STS is #19, an alternative method is given by means of the 1-factorizations of the complete graph K8 associated to the two KTSs.Moreover, we present three new visual solutions to the schoolgirl problem, and we catalogue most of the classical (or interesting) solutions in the literature in terms of what KTS(15)s they are isomorphic to.This paper provides background on a classical topic, while shedding new light on the problem as well.

Total dominator chromatic number of Kneser graphs

Decomposition into special substructures inheriting significant properties is an important method for the investigation of some mathematical structures. A total dominator coloring (briefly, a TDC) of a graph G is a proper coloring (i.e. a partition of the vertex set V(G) into independent subsets named color classes) in which each vertex of the graph is adjacent to all of vertices of some color class. The total dominator chromatic number of G is the minimum number of color classes in a TDC of G. In this paper among some other results and by using the existence of Steiner triple systems, we determine the total dominator chromatic number of the Kneser graph for each

Betweenness structures of small linear co-size

One way to study the combinatorics of finite metric spaces is to study the betweenness relation associated with the metric space. In the hypergraph metrization problem, one has to find and characterize metric betweennesses with collinear triples (or alternatively, non-degenerate triangles) that coincide with the edges of a given 3-uniform hypergraph. Metrizability of different kinds of hypergraphs was investigated in the last decades. Chen showed that Steiner triple systems are not metric, while Richmond and Richmond characterized linear betweennesses, i.e. metric betweennesses that realize the complete 3-uniform hypergraph. The latter result was also generalized to pseudometric (almost-metric) betweennesses by Beaudou et al. In this paper, we further extend this theory by characterizing the largest non-linear almost-metric betweennesses that satisfy certain hereditary properties, as well as the ones that contain a small linear number of non-degenerate triangles.

On asymptotic packing of geometric graphs

A set of geometric graphs is geometric-packable if it can be asymptotically packed into every sequence of drawings of the complete graph K n . For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When G is the 4-cycle (or 4-cycle with a chord), we show that the set of plane drawings of G is geometric-packable. In contrast, the analogous statement is false when G is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is convex-packable if it can be asymptotically packed into the convex drawings of the complete graphs. For each planar Hamiltonian graph G , we determine whether or not a plane G is convex-packable. Many of our proofs explicitly construct these packings; in these cases, the packings exhibit a symmetry that mirrors the vertex transitivity of K n .

Automorphism groups of Steiner triple systems

If $G$ is a finite group then there is an integer $M_G$ such that$,$ for $u\ge M_G$ and $u\equiv 1$ or $3$ (mod 6), there is a Steiner triple system $U$ on $u$ points for which ${\rm Aut} U \cong G. \ $ If $V$ is a Steiner triple system then there is an integer $N_V$ such that$,$ for $u\ge N_V$ and $u\equiv 1$ or $3$ $($mod $6),$ there is a Steiner triple system $U$ on $u$ points having $V$ as an ${\rm Aut} U$-invariant subsystem such that ${\rm Aut} U\cong{\rm Aut} V $ and ${\rm Aut} U $ induces ${\rm Aut} V$ on $V$.

Novel space shift keying MIMO technique based on a Steiner triple system

A novel space shift keying (SSK) multiple–input multiple–output (MIMO) technique based on Steiner triple system is proposed and analyzed in this article. SSK attracted considerable research interest in the past few years driven by the several promised inherent advantages including low error probability, low computational complexity and a very simple hardware implementation with very low cost and power consumption. Yet, the spectral efficiency of SSK increases with a base two logarithm of the number of transmit antennas and high data rates are only viable with a massive and impractical number of transmit antennas. Alternatively, generalized SSK (GSSK) scheme is considered, where a combination of antennas is activated at each time instant. GSSK promises the use of arbitrary number of transmit antennas not necessarily a power of two integer. Also, GSSK can attain high data rate with low number of transmit antennas at the cost of substantial degradation in the error performance. In this study, a Steiner triple system is utilized to propose a tailored SSK scheme with substantial reduction in the required number of transmit antennas, without compromising the error probability. It is shown that the proposed Steiner–SSK (S–SSK) MIMO system achieves almost identical error performance to a conventional SSK system but with nearly 90% reduction in the number of transmit antennas. As well, the average bit error rate (ABER) of S–SSK is shown to outperform GSSK by at least 3dB. It is also reported that a S–SSK system accomplishes significant reduction in hardware cost, power consumption, and computational complexity as compared to conventional SSK scheme. Yet, GSSK is shown to marginally outperforms S–SSK in these metrics as it requires smaller number of transmit antennas per a target spectral efficiency.

Block avoiding point sequencings of partial Steiner systems

A partial(n,k,t)_lambda -system is a pair (X,{mathcal {B}}) where X is an n-set of vertices and {mathcal {B}} is a collection of k-subsets of X called blocks such that each t-set of vertices is a subset of at most lambda blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of {0,ldots ,n-1}. A sequencing is ell -block avoiding or, more briefly, ell -good if no block is contained in a set of ell vertices with consecutive labels. Here we give a short proof that, for fixed k, t and lambda , any partial (n,k,t)_lambda -system has an ell -good sequencing for some ell =Theta (n^{1/t}) as n becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case k=t+1 where results of Kostochka, Mubayi and Verstraëte show that the value of ell cannot be increased beyond Theta ((n log n)^{1/t}). A special case of our result shows that every partial Steiner triple system (partial (n,3,2)_1-system) has an ell -good sequencing for each positive integer ell leqslant 0.0908,n^{1/2}.

Every Steiner Triple System Contains Almost Spanning $d$-Ary Hypertree

In this paper we make a partial progress on the following conjecture: for every $\mu>0$ and large enough $n$, every Steiner triple system $S$ on at least $(1+\mu)n$ vertices contains every hypertree $T$ on $n$ vertices. We prove that the conjecture holds if $T$ is a perfect $d$-ary hypertree.

Hypergraphs without non-trivial intersecting subgraphs

AbstractA hypergraph $\mathcal{F}$ is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of $\mathcal{F}$ . Mubayi and Verstraëte showed that for every $k \ge d+1 \ge 3$ and $n \ge (d+1)k/d$ every $k$ -graph $\mathcal{H}$ on $n$ vertices without a non-trivial intersecting subgraph of size $d+1$ contains at most $\binom{n-1}{k-1}$ edges. They conjectured that the same conclusion holds for all $d \ge k \ge 4$ and sufficiently large $n$ . We confirm their conjecture by proving a stronger statement.They also conjectured that for $m \ge 4$ and sufficiently large $n$ the maximum size of a $3$ -graph on $n$ vertices without a non-trivial intersecting subgraph of size $3m+1$ is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.