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In [3], the authors adapted the edge-graceful graph labeling definition into block designs. In this article, we adapt the line-graceful graph labeling definition into block designs and define a block design (V; B) with |V| = ν as line-graceful if there exists a function f: B → {0; 1;: ::; v - 1} such that the induced mapping f+: V → Zv given by f+(x) = ∑A∊B: xA∊BA f(A) (mod v) is a bijection. In this article, the cases that are incomplete in terms of block-graceful labelings, are completed in terms of line-graceful labelings. Moreover, we prove that there exists a line-graceful Steiner quadruple system of order 2n for all n ≤ 3 by using a recursive construction. [ABSTRACT FROM AUTHOR]

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A 3-way (v , k , t) trade T of volume m consists of three pairwise disjoint collections T 1 , T 2 and T 3 , each of m blocks of size k , such that for every t -subset of v -set V , the number of blocks containing this t -subset is the same in each T i for 1 ≤ i ≤ 3 . If any t -subset of found(T ) occurs at most once in each T i for 1 ≤ i ≤ 3 , then T is called 3-way (v , k , t) Steiner trade. We attempt to complete the spectrum S 3 s (v , k) , the set of all possible volume sizes, for 3-way (v , k , 2) Steiner trades, by applying some block designs, such as BIBDs, RBs, GDDs, RGDDs, and r × s packing grid blocks. Previously, we obtained some results about the existence some 3-way (v , k , 2) Steiner trades. In particular, we proved that there exists a 3-way (v , k , 2) Steiner trade of volume m when 12 (k − 1) ≤ m for 15 ≤ k (Rashidi and Soltankhah in Discrete Math. 339(12): 2955–2963, 2016). Now, we show that the claim is correct also for k ≤ 14 . [ABSTRACT FROM AUTHOR]

We propose a method of constructing block designs which combine genetic algorithm and a method for constructing designs with a prescribed automorphism group using tactical decompositions (i.e., orbit matrices). We apply this method to construct new Steiner systems with parameters S(2,5,45) $S(2,5,45)$ and new symmetric designs with parameters (71, 15, 3). [ABSTRACT FROM AUTHOR]

A bipartite biregular (m,n;g) $(m,n;g)$‐graph Γ ${\rm{\Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} $\{m,n\}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$‐graph of minimum order. In their 2019 paper, Filipovski, Ramos‐Rivera, and Jajcay present lower bounds on the orders of bipartite biregular (m,n;g) $(m,n;g)$‐graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well‐known classical results relating the existence of k $k$‐regular Moore graphs of even girths g=6,8 $g=6,8$, and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an S(2,k,v) $S(2,k,v)$‐Steiner system yields the existence of a bipartite biregular k,v−1k−1;6 $\left(k,\frac{v-1}{k-1};6\right)$‐cage, and, vice versa, the existence of a bipartite biregular (k,n;6) $(k,n;6)$‐cage whose order is equal to one of our lower bounds yields the existence of an S(2,k,1+n(k−1)) $S(2,k,1+n(k-1))$‐Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of (3,n;6) $(3,n;6)$‐bipartite biregular cages for all integers n≥4 $n\ge 4$. Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of (n+1,n2+1;8) $(n+1,{n}^{2}+1;8)$‐, (n2+1,n3+1;8) $({n}^{2}+1,{n}^{3}+1;8)$‐, (n,n+2;8) $(n,n+2;8)$‐, (n+1,n3+1;12) $(n+1,{n}^{3}+1;12)$‐ and (n+1,n2+1;16) $(n+1,{n}^{2}+1;16)$‐bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14. [ABSTRACT FROM AUTHOR]

For an ordering of the blocks of a design, the point sum of an element is the sum of the indices of blocks containing that element. Block labelling for popularity asks for the point sums to be as equal as possible. For Steiner systems of order v and strength t in general, the average point sum is O (v 2 t - 1) ; under various restrictions on block partitions of the Steiner system, the difference between the largest and smallest point sums is shown to be O (v (t + 1) / 2 log v) . Indeed for Steiner triple systems, direct and recursive constructions are given to establish that systems exist with all point sums equal for more than two thirds of the admissible orders. [ABSTRACT FROM AUTHOR]

Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs, whose blocks are of size k taken from an n-set, is a partition of all the k-subsets of the n-set into disjoint copies of block designs, defined on the n-set, and with the same parameters. The current most intriguing question in this direction is whether large sets of Steiner quadruple systems exist and to provide explicit constructions for those parameters for which they exist. In view of its difficulty no one ever presented an explicit construction even for one nontrivial order. Hence, we seek for related generalizations. As generalizations, to the existence question of large sets, we consider two related questions. The first one is to provide constructions for sets on Steiner systems in which each block (quadruple or a k-subset) is contained in exactly μ systems. The constructions of such systems also yield secure protocols for the generalized Russian cards problem. The second question is to provide constructions for large set of H-designs (mainly for quadruples, but also for larger block size), which have applications in threshold schemes and in quantum jump codes. We prove the existence of such systems for many parameters using orthogonal arrays, perpendicular arrays, ordered designs, sets of permutations, and one-factorizations of the complete graph. [ABSTRACT FROM AUTHOR]

Let n and ℓ be positive integers. Recent papers by Kreher, Stinson, and Veitch have explored variants of the problem of ordering the points in a triple system (such as a Steiner triple system [STS], directed triple system, or Mendelsohn triple system) on n points so that no block occurs in a segment of ℓ consecutive entries (thus the ordering is locally block‐avoiding). We describe a greedy algorithm which shows that such an ordering exists, provided that n is sufficiently large when compared with ℓ. This algorithm leads to improved bounds on the number of points in cases where this was known, but also extends the results to a significantly more general setting (which includes, e.g., orderings that avoid the blocks of a design). Similar results for a cyclic variant of this situation are also established. We construct STSs and quadruple systems where ℓ can be large, showing that a bound of Stinson and Veitch is reasonable. Moreover, we generalize the Stinson–Veitch bound to a wider class of block designs and to the cyclic case. The results of Kreher, Stinson, and Veitch were originally inspired by results of Alspach, Kreher, and Pastine, who (motivated by zero‐sum avoiding sequences in abelian groups) were interested in orderings of points in a partial Steiner triple system where no segment is a union of disjoint blocks. Alspach et al. show that, when the system contains at most k pairwise disjoint blocks, an ordering exists when the number of points is more than 15k−5. By making use of a greedy approach, the paper improves this bound to 9k+O(k2∕3). [ABSTRACT FROM AUTHOR]

We study the localization number of incidence graphs of designs. In the localization game played on a graph, the cops attempt to determine the location of an invisible robber via distance probes. The localization number of a graph G, written ζ(G), is the minimum number of cops needed to ensure the robber's capture. We present bounds on the localization number of incidence graphs of balanced incomplete block designs. Exact values of the localization number are given for the incidence graphs of projective and affine planes. Bounds are given for Steiner systems and for transversal designs. [ABSTRACT FROM AUTHOR]

A graph is n-existentially closed (n-e.c.) if for any disjoint subsets A, B of vertices with A ∪ B = n , there is a vertex z ∉ A ∪ B adjacent to every vertex of A and no vertex of B. For a block design with block set B , its block intersection graph is the graph whose vertex set is B and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be 2-e.c. In particular, we study the λ -fold triple systems with λ ≥ 2 and determine for which parameters their block intersection graphs are 1- or 2-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called { 1 } -block intersection graphs are investigated, and the necessary and sufficient conditions for such graphs to be 2-e.c. are established. [ABSTRACT FROM AUTHOR]

Let v ≡ 4 (mod 6) and v ≥ 4 . A canonical Kirkman packing design of order v, denoted by CKPD (v) , is a resolvable packing with r = (v - 4) / 2 parallel classes such that (i) each parallel class consists of a size 4 block and (v - 4) / 3 triples; (ii) the leave consists of the union of (v - 4) / 2 vertex-disjoint edges and a K 4 with no vertices in common with those edges. A canonical Kirkman packing design is said to be doubly resolvable if there exist a pair of orthogonal resolutions. A doubly resolvable packing design is the generalization of the Kirkman square, which is inextricably bound up with the existence of some constant weight codes such as constant composition codes and permutation codes, etc. In this paper, we establish the spectra of doubly resolvable CKPD (v) s with 36 possible exceptions for v. As its direct application, a class of permutation codes are obtained. [ABSTRACT FROM AUTHOR]

Pairwise Balanced Design (PBD) is a pair (X, Ɓ) where X is a set of treatments and Ɓ is a collection of subsets of X called blocks, such that each pair of treatments is contained in precisely one block. PBD plays important role in design theories, it is used to construct other important designs such as Steiner Triple System (STS). A Steiner triple system is an ordered pair (X, Ɓ), where X is a finite set of points (Treatments) and B is a set of all 3-element subsets of X called triples, such that each pair of distinct elements of X occurs together in exactly one triple of B. The research work aims at applying a class of PBD(n, K, 1) when K ={3, 4} and λ = 1 to construct STS(2n +1). Theorem was proposed and proved and a certain inequality was derived as condition which must be satisfied for the construction to hold. Thus, for all n ≡ 1, 3(mod 3) of any PBD(n, {3, 4}) there exists an STS(2n +1) provided n ≥ l(s -1) + 1, where l is the size of the largest block of the PBD and s is the size of the smallest block of the PBD. Hence, STS(21) was constructed from a PBD(10, {3, 4}, 1). [ABSTRACT FROM AUTHOR]

To every Steiner system with parameters (t,k,v) on {1,...,v} and blocks B, we can assign its characteristic vector χB, which is a (0,1)‐vector whose entries are indexed by the k‐subsets of {1,...,v} such that for each k‐subset A of {1,...,v},χB(A)=1 if and only if A∈B. In this paper, we show that the dimension of the vector space generated by all of the characteristic vectors of Steiner systems with parameters (t,k,v) is (vk)−(vt)+1, provided that 1≤t≤k≤v/2 and there is at least one such system. [ABSTRACT FROM AUTHOR]

A μ-way (v, k, t) trade is a pair T=(X,{T1,T2,...,Tμ}) such that for eacht-subset of v-set X the number of blocks containing this t-subset is the same in each Ti(1≤i≤μ). In the other words for each 1≤i

A decomposition of the blocks of an STS(v) into partial parallel classes of size m is equivalent to a Kirkman signal set KSS(v,m). We give decompositions of STS(4v-3) into classes of size v-1 when v≡3(mod6), v≠3. We also give decompositions of STS(v) into classes of various sizes when v is a product of two arbitrary integers that are both congruent to 3(mod6). These results produce new families of KSS(v,m). [ABSTRACT FROM AUTHOR]

We introduce m-near-resolvable block designs. We establish a correspondence between such block designs and a subclass of (optimal equidistant) q-ary constant-weight codes meeting the Johnson bound. We present constructions of m-near-resolvable block designs, in particular based on Steiner systems and super-simple t-designs. [ABSTRACT FROM AUTHOR]

This work presents a recursive construction for simple t-designs using resolutions of the ingredient designs. The result extends a construction of t-designs in our recent paper van Trung (Des Codes Cryptogr 83:493-502, 2017). Essentially, the method in van Trung (Des Codes Cryptogr 83:493-502, 2017) describes the blocks of a constructed design as a collection of block unions from a number of appropriate pairs of disjoint ingredient designs. Now, if some pairs of these ingredient t-designs have both suitable s-resolutions, then we can define a distance mapping on their resolution classes. Using this mapping enables us to have more possibilities for forming blocks from those pairs. The method makes it possible for constructing many new simple t-designs. We give some application results of the new construction. [ABSTRACT FROM AUTHOR]

The unique Steiner triple system of order 7 has a point-block incidence graph known as the Heawood graph. Motivated by questions in combinatorial matrix theory, we consider the problem of constructing a faithful orthogonal representation of this graph, i.e., an assignment of a vector in C d to each vertex such that two vertices are adjacent precisely when assigned nonorthogonal vectors. We show that d = 10 is the smallest number of dimensions in which such a representation exists, a value known as the minimum semidefinite rank of the graph, and give such a representation in 10 real dimensions. We then show how the same approach gives a lower bound on this parameter for the incidence graph of any Steiner triple system, and highlight some questions concerning the general upper bound. [ABSTRACT FROM AUTHOR]

Let Σ = (X, B) be a 6-cycle system of order v, so v = 1, 9 mod 12. A c-colouring of type s is a map φ B → C, with C set of colours, such that exactly c colours are used and for every vertex x all the blocks containing x are coloured exactly with s colours. Let v-1/2 = qs + r, with q, r ≥ 0. φ is equitable if for every vertex x the set of the v-1/2 blocks containing x is partitioned in r colour classes of cardinality q + 1 and s - r colour classes of cardinality q. In this paper we study bicolourings and tricolourings, for which, respectively, s = 2 and s = 3, distinguishing the cases v = 12k+1 and v = 12k+9. In particular, we settle completely the case of s = 2, while for s = 3 we determine upper and lower bounds for c. [ABSTRACT FROM AUTHOR]

In the Steiner Tree problem one is given an undirected graph, a subset T of its vertices, and an integer k and the question is whether there is a connected subgraph of the given graph containing all the vertices of T and at most k other vertices. The vertices in the subset T are called terminals and the other vertices are called Steiner vertices. Recently, Pilipczuk et al. (55th IEEE Annual Symposium on Foundations of Computer Science, FOCS, 2014) gave a polynomial kernel for Steiner Tree in planar graphs and graphs of bounded genus, when parameterized by $$|T|+k$$ , the total number of vertices in the constructed subgraph. In this paper we present several polynomial time applicable reduction rules for Steiner Tree in graphs of bounded genus. In an instance reduced with respect to the presented reduction rules, the number of terminals | T| is at most cubic in the number of other vertices k in the subgraph. Hence, using and improving the result of Pilipczuk et al., we give a polynomial kernel for Steiner Tree in graphs of bounded genus for the parameterization by the number k of Steiner vertices in the solution. We give better bounds for Steiner Tree in planar graphs. [ABSTRACT FROM AUTHOR]

Research on the existence of large sets of Kirkman triple systems (LKTS) extends from the mid-eighteen hundreds to the present. In this paper we review known direct approaches of constructing LKTS and present new ideas of direct constructions. We finally prove the existence of an LKTS( v) where $$v \in \{69,141,165,213,285,309,333\}$$ . Combining recursive constructions yields several new infinite classes. [ABSTRACT FROM AUTHOR]

A (v, 3,2)-covering is a family of 3-subsets of a v-set, called blocks, such that any two elements of v-set appear in at least one of the blocks. In this paper, we propose new construction of (v,3,2)-coverings with the minimum number of blocks. This construction represents a generalization of Bose's and Skolem's constructions of Steiner systems S(2,3,6n + 3) and S(2,3,6n + 1). Unlike the existing constructions, our construction is direct and it uses the set of base blocks and permutation p, so by applying it to the remaining blocks of (v, 3,2)-coverings are obtained. [ABSTRACT FROM AUTHOR]

This article investigates the robustness of binary incomplete block designs against giving rise to a disconnected design in the event of observation loss. A link is established between theE-value of a planned design and the extent of observation loss that can be experienced while still guaranteeing an eventual design from which all treatment contrasts can be estimated. Patterns of missing observations covered include loss of entire blocks and loss of individual observations. Simple bounds are provided enabling practitioners to easily assess the robustness of a planned design. [ABSTRACT FROM AUTHOR]

Let 픽qn be a vector space of dimension n over the finite field ---q. A q-analog of a Steiner system (also known as a q-Steiner system), denoted Sq (t, k, n), is a set S of k-dimensional subspaces of ---qn such that each t-dimensional subspace of ---qn is contained in exactly one element of S. Presently, q-Steiner systems are known only for t = 1, and in the trivial cases t = k and k = n. In this paper, the first nontrivial q-Steiner systems with t ≥ 2 are constructed. Specifically, several nonisomorphic q-Steiner systems S2 (2, 3, 13) are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of GL(13, 2). This approach leads to an instance of the exact cover problem, which turns out to have many solutions. [ABSTRACT FROM AUTHOR]

A gradient-constrained discounted Steiner tree is a network interconnecting given set of nodes in Euclidean space where the gradients of the edges are all no more than an upper bound which defines the maximum gradient. In such a tree, the costs are associated with its edges and values are associated with nodes and are discounted over time. In this paper, we study the problem of optimally locating a single Steiner point in the presence of the gradient constraint in a tree so as to maximize the sum of all the discounted cash flows, known as the net present value (NPV). An edge in the tree is labelled as a b edge, or a m edge, or an f edge if the gradient between its endpoints is greater than, or equal to, or less than the maximum gradient respectively. The set of edge labels at a discounted Steiner point is called its labelling. The optimal location of the discounted Steiner point is obtained for the labellings that can occur in a gradient-constrained discounted Steiner tree. In this paper, we propose the gradient-constrained discounted Steiner point algorithm to optimally locate the discounted Steiner point in the presence of a gradient constraint in a network. This algorithm is applied to a case study. This problem occurs in underground mining, where we focus on the optimization of underground mine access to obtain maximum NPV in the presence of a gradient constraint. The gradient constraint defines the navigability conditions for trucks along the underground tunnels. [ABSTRACT FROM AUTHOR]

A gradient-constrained discounted Steiner tree T is a maximum Net Present Value (NPV) tree, spanning a given set N of nodes in space with edges whose gradients are all no more than an upper bound m which is the maximum gradient. The nodes in T but not in N are referred to as discounted Steiner points. Such a tree has costs associated with its edges and values associated with its nodes. In order to reach the nodes in the tree, the edges need to be constructed. The edges are constructed in a particular order and the costs of constructing the edges and the values at the nodes are discounted over time. In this paper, we study the optimal tree configurations so as to maximize the sum of all the discounted cash flows, known as the NPV. An application of this problem occurs in underground mining, where we want to optimally locate a junction point in the underground access network to maximize the NPV in the presence of the gradient constraint. This constraint defines the navigability conditions on mining vehicles along the underground tunnels. Labellings are essential for defining a tree configuration and indicate gradients on the edges of the network. An edge in a gradient-constrained discounted Steiner tree is labelled as an f edge, an m edge or a b edge, if the gradient is less, equal or greater than m respectively. Each tree configuration is identified by the labellings of its edges. In this paper the non-optimal sets of labellings of edges that are incident with the discounted Steiner point in a gradient-constrained discounted Steiner network are classified. This reduces the number of configurations that need to be considered when optimizing. In addition, the gradient-constrained discounted Steiner point algorithm is outlined. [ABSTRACT FROM AUTHOR]

Here we study the automorphism groups of $$1$$ -designs constructed from finite nonabelian simple groups by using two methods presented in Moori (Information Security, Coding Theory and Related Combinatorics, ). We obtain some general results for both and improve one of these methods. In an application to the sporadic Mathieu groups $$M_{n}$$ , we are able to retrieve the Steiner systems $$S(t,t+3,n)$$ where $$(n,t)\in \{(22,3),(23,4),(24,5)\}$$ . [ABSTRACT FROM AUTHOR]

The expansion G+ of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a new vertex disjoint from V(G) such that distinct edges are enlarged by distinct vertices. Let ex3 (n, F) denote the maximum number of edges in a 3-uniform hypergraph with n vertices not containing any copy of a 3-uniform hypergraph F. The study of ex3(n, G+) includes some well-researched problems, including the case that F consists of k disjoint edges, G is a triangle, G is a path or cycle, and G is a tree. In this paper we initiate a broader study of the behavior of ex3(n, G+). Specifically, we show ex3 (n, K s,t+) = Θ(n3-3/s) whenever t > (s -- 1)! and s ≥ 3. One of the main open problems is to determine for which graphs G the quantity ex3 (n, G+) is quadratic in n. We show that this occurs when G is any bipartite graph with Turan number o(nφ) where φ = 1+√5/2, and in particular this shows ex3 (n, G+) = O(n²) when G is the three-dimensional cube graph. [ABSTRACT FROM AUTHOR]

A μ-way (v, k, t) trade of volume m consists of μ disjoint collections T1, T2, . . . Tμ, each of m blocks, such that for every t-subset of v-set V the number of blocks containing this t-subset is the same in each Ti (1 ≤ i ≤ μ). In other words any pair of collections {Ti, Tj}, 1 ≤ i < j ≤ μ is a (v, k, t) trade of volume m. In this paper we investigate the existence of ≥-way (v, k, t) trades and prove the existence of. (i) 3-way (v, k, 1) trades Steiner trades) of each volume m,m ≥ 2. (ii) 3-way (v, k, 2) trades of each volume m,m ≥ 6 except possibly m = 7. We establish the non-existence of 3-way (v, 3, 2) trade of volume 7. It is shown that the volume of a 3-way (v, k, 2) Steiner trade is at least 2k for k ≥ 4. Also the spectrum of 3-way (v, k, 2) Steiner trades for k = 3 and 4 are specified. [ABSTRACT FROM AUTHOR]

A criterion for the continuity of the Steiner ratio, the Steiner subratio, and the Steiner-Gromov ratio in the space of all compact metric spaces with the Gromov-Hausdorff metric is obtained. It is also proved that these functions are upper semicontinuous. [ABSTRACT FROM AUTHOR]

In 2009 it was established that any partial Steiner triple system of order u has an embedding of order v for each v &#8805: 2u+1 such that v = 1, 3 (mod 6), in accordance with a conjecture of Lindner. It is known that for each u ≥ 9, there exists a partial Steiner triple system of order u that does not have an embedding of order v for any v < 2u + 1, so this result is best possible in one sense. Many partial Steiner triple systems do have embeddings of orders smaller than 2u + 1, however, although little is known about when such embeddings exist. In this paper we construct embeddings of orders less than 2u+1 for partial Steiner triple systems with few triples. In particular, we show that a partial Steiner triple system of order u ≥ 62 with at most u²/50-11u/100-116/75 triples has an embedding of order v for each admissible integer v ≥ 8u+17/5. [ABSTRACT FROM AUTHOR]

We prove that for all n ≡ 13 or 37 (mod 72), there exists a biembedding of a pair of Steiner triple systems of order n in an orientable pseudosurface having precisely one regular pinch point of multiplicity 2. [ABSTRACT FROM AUTHOR]

Ordering the blocks of a design, the point sum of an element is the sum of the indices of blocks containing that element. Block labelling for popularity asks for the point sums to be as equal as possible. When all point sums are equal, the system is egalitarian; when point sums differ by at most one, it is almost egalitarian. For Steiner quadruple systems, a doubling construction is adapted to establish that an egalitarian S (3 , 4 , v) exists whenever v ≡ 4 , 20 (mod 24) and that an almost egalitarian S (3 , 4 , v) exists whenever v ≡ 8 , 16 (mod 24) and v ≠ 8. [ABSTRACT FROM AUTHOR]

We prove that, for even m, the chromatic index of the projective triple system STS( [ABSTRACT FROM AUTHOR]

The proliferating innovative ways of promoting brands have necessitated the assessment of the effectiveness of the tool being used. As celebrity endorsement is believed to have the stopping power in the cluttered media environment, the effectiveness of which should be assessed. While there are many ways to assess the celebrity endorsement, this research paper examines the effectiveness of celebrity used in endorsements with a different approach. This paper introduces a new recall concept termed as celebrityaided brand recall and brand-aided celebrity recall. It also identifies the selected celebrity-aided recall of brands and selected brand-aided recall of celebrity. Further, it examines the extent to which the celebrity influenced the respondents using the Hierarchy-of-Effects Model of Lavidge and Steiner. [ABSTRACT FROM AUTHOR]

Necessary conditions for existence of a resolvable group divisible design (GDD) with block size 3 and type [ABSTRACT FROM AUTHOR]

For a block design $${\mathcal{D}}$$ , a series of block intersection graphs G, or i-BIG( $${\mathcal{D}}$$), i = 0, . . . , k is defined in which the vertices are the blocks of $${\mathcal{D}}$$ , with two vertices adjacent if and only if the corresponding blocks intersect in exactly i elements. A silver graph G is defined with respect to a maximum independent set of G, called an α- set. Let G be an r-regular graph and c be a proper ( r + 1)-coloring of G. A vertex x in G is said to be rainbow with respect to c if every color appears in the closed neighborhood $${N[x] = N(x) \cup \{x\}}$$ . Given an α-set I of G, a coloring c is said to be silver with respect to I if every $${x\in I}$$ is rainbow with respect to c. We say G is silver if it admits a silver coloring with respect to some I. Finding silver graphs is of interest, for a motivation and progress in silver graphs see Ghebleh et al. (Graphs Combin 24(5):429-442, ) and Mahdian and Mahmoodian (Bull Inst Combin Appl 28:48-54, ). We investigate conditions for 0-BIG( $${\mathcal{D}}$$) and 1-BIG( $${\mathcal{D}}$$) of Steiner 2-designs $${{\mathcal{D}}=S(2,k,v)}$$ to be silver. [ABSTRACT FROM AUTHOR]

It is well known that when [ABSTRACT FROM AUTHOR]

Abstract: A flower, , around a point in a Steiner triple system is the set of all triples in which contain the point , namely . This paper determines the possible number of common flowers that two Steiner triple systems can have in common. For all admissible pairs where we construct a pair of Steiner triple systems of order where the flowers around elements of are identical in both Steiner triple systems, except for the pairs , and . Equivalently this result shows that there is a Steiner triple trade of foundation that can be embedded in a for each admissible and except when or . [Copyright &y& Elsevier]

In this paper, an electromagnetism-like approach (EM) for solving the maximum set splitting problem (MSSP) is applied. Hybrid approach consisting of the movement based on the attraction-repulsion mechanisms combined with the proposed scaling technique directs EM to promising search regions. Fast implementation of the local search procedure additionally improves the efficiency of overall EM system. The performance of the proposed EM approach is evaluated on two classes of instances from the literature: minimum hitting set and Steiner triple systems. The results show, except in one case, that EM reaches optimal solutions up to 500 elements and 50000 subsets on minimum hitting set instances. It also reaches all optimal/best-known solutions for Steiner triple systems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the problem of finding the largest integer for which there exists a set of points and blocks in a balanced incomplete block design such that none of the points lie on any of the blocks. We investigate this problem for two types of BIBDs: projective planes and Steiner triple systems. For a Steiner triple system on points, we prove that , and we determine necessary and sufficient conditions for equality to be attained in this bound. For a projective plane of order , we prove that , and we show that equality can be attained in this bound whenever is an even power of two. [Copyright &y& Elsevier]

Let X be a v-set [ABSTRACT FROM AUTHOR]

A Skolem-type sequence is a sequence ( s, . . . , s) of positive integers $${i\in D}$$ such that for each $${i\in D}$$ there is exactly one $${j\in \{1, \ldots , t - i\}}$$ such that s = s = i. Positions in the sequence not occupied by integers $${i\in D}$$ contain null elements. In 1939, Peltesohn solved the existence problem for cyclic Steiner triple systems for v ≡ 1, 3(mod 6), v ≠ 9. Using the same technique in 1981, Colbourn and Colbourn extended the solution to all admissible λ > 1. It is known that Skolem-type sequences may be used to construct cyclic Steiner triple systems as well as cyclic triple systems with λ = 2. The main result of this paper is an extension of former results to cyclic triple systems with λ > 2. In addition we introduce a new kind of Skolem-type sequence. [ABSTRACT FROM AUTHOR]

Abstract: A Steiner system is a pair , where is a -element set and is a set of -subsets of , called blocks, with the property that every -element subset of is contained in a unique block. The sub-design in a Steiner quadruple system is said to be a spanning block design. The order of a Steiner quadruple system with a spanning block design should satisfy the necessary condition . It is proved that the above necessary condition is also sufficient. As a consequence, it is also proved that a 3-BD exists for any . [Copyright &y& Elsevier]

Higman asked which block graphs of Steiner triple systems of order v satisfy the 4-vertex condition and left the cases v = 9, 13, 25 unsettled.We give a complete answer to this question by showing that the affine plane of order 3 and the binary projective spaces are the only such systems. The major part of the proof is to show that no block graph of a Steiner triple system of order 25 satisfies the 4-vertex condition. [ABSTRACT FROM AUTHOR]

The article presents information on a research paper on discrete mathematics which concludes that two fold systems hold good for all natural numbers. The authors construct a pairwise balanced design (PBD) for the order 6n+4 and two fold exit systems for the order 3n+2. It makes use of Steiner Triple System (STS), a type of block design in combinatorial mathematics, already existing for the order 6n+1 and 6n+3 and PBD for the order 6n+5.

The original classiffcation of PBIBDs defined group divisible designs GDD(v=v1+ v2+…+vg,g,k,λ1,λ2) with λ1≠0. In this paper, we prove that the necessary conditions are suffcient for the existence of the group divisible designs with two groups of unequal sizes and block size three with λ2=1 [ABSTRACT FROM AUTHOR]

n H-design is said to be (1, α)-resolvable, if its block set can be partitioned into α-parallel classes, each of which contains every point of the design exactly α times. When α = 1, a (1, α)-resolvable H-design of type g is simply called a resolvable H-design and denoted by RH( g), for which the general existence problem has been determined leaving mainly the case of g ≡ 0 (mod 12) open. When α = 2, a (1, 2)-RH(1) is usually called a (1, 2)-resolvable Steiner quadruple system of order n, for which the existence problem is far from complete. In this paper, we consider these two outstanding problems. First, we prove that an RH(12) exists for all n ≥ 4 with a small number of possible exceptions. Next, we give a near complete solution to the existence problem of (1, 2)-resolvable H-designs with group size 2. As a consequence, we obtain a near complete solution to the above two open problems. [ABSTRACT FROM AUTHOR]

A Steiner System, denoted S(t, k, υ), is a vertex set X containing υ vertices, and a collection of subsets of X of size k, called blocks, such that every t vertices from X are in exactly one of the blocks. A Steiner Triple System, or STS, is a special case of a Steiner System where t = 2, k = 3 and υ = 1 or 3 (mod6) [7]. A Bi-Steiner Triple System, or BSTS, is a Steiner Triple System with the vertices colored in such a way that each block of vertices receives precisely two colors. Out of the 80 BSTS(15)s, only 23 are colorable [1]. In this paper, using a computer program that we wrote, we give a complete description of all proper colorings, all feasible partitions, chromatic polynomial and chromatic spectrum of every colorable BSTS(15). [ABSTRACT FROM AUTHOR]