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Cameron defined the concept of generalized t $t$‐designs, which generalized t $t$‐designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized t $t$‐designs and group divisible t $t$‐designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible t $t$‐designs, and then point out close connections with various well‐known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible t $t$‐designs and almost completely determine their existence for t=2,3 $t=2,3$ and small block sizes. [ABSTRACT FROM AUTHOR]

We show that the maximum number of columns in E(s2) $\,E({s}^{2})$‐optimal supersaturated designs (SSDs) with 16 rows and smax=4 ${s}_{{\rm{\max }}}=4$ is 60 by showing that there exists no resolvable 2‐(16, 8, 35) design such that any two blocks from different parallel classes intersect in 3, 5, or 4 points. This is accomplished by an exhaustive computer search that uses the parallel class intersection pattern method to reduce the search space. We also classify all nonisomorphic E(s2) $\,E({s}^{2})$‐optimal 5‐circulant SSDs with 16 rows and smax=8 ${s}_{{\rm{\max }}}=8$. [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]

Decomposable designs are important combinatorial designs, and have applications in constant weight codes, perfect threshold schemes and graph decompositions. Let λ $\lambda $ be an even positive integer. A λ2 $\frac{\lambda }{2}$‐decomposable super‐simple (v,k,λ) $(v,k,\lambda)$‐balanced incomplete block design (BIBD) is a super‐simple (v,k,λ) $(v,k,\lambda)$‐BIBD, (V,ℬ) $(V,{\rm{ {\mathcal B} }})$, where ℬ=ℬ1∪ℬ2 ${\rm{ {\mathcal B} }}={{\rm{ {\mathcal B} }}}_{1}\cup {{\rm{ {\mathcal B} }}}_{2}$, and each (V,ℬi) $(V,{{\rm{ {\mathcal B} }}}_{i})$ is a (v,k,λ2) $(v,k,\frac{\lambda }{2})$‐BIBDs for i=1,2 $i=1,2$. In this paper, for λ∈{4,6} $\lambda \in \{4,6\}$, it is proved that the necessary conditions for the existence of a λ2 $\frac{\lambda }{2}$‐decomposable super‐simple (v,4,λ) $(v,4,\lambda)$‐BIBD are also sufficient. [ABSTRACT FROM AUTHOR]

In this paper we provide a 4‐GDD of type 2255 ${2}^{2}{5}^{5}$, thereby solving the existence question for the last remaining feasible type for a 4‐GDD with no more than 30 points. We then show that 4‐GDDs of type 2t5s ${2}^{t}{5}^{s}$ exist for all but a finite specified set of feasible pairs (t,s) $(t,s)$. [ABSTRACT FROM AUTHOR]

This paper addresses the question of how intercalates occur in the two known infinite families of triple arrays, the Paley triple arrays constructed in 2005 by Preece et al., and the Triple arrays from difference sets in 2017 by Nilson and Cameron. The main reason for doing this is that the number of such embedded Latin squares is often used when checking whether two arrays are isotopic or not. We determine sharp bounds for the number and density of intercalates for the main subclasses of these families respectively. We also prove the existence of an infinite family of triple arrays in which every two occurrences of an entry lie in an intercalate. [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]

In this paper, we focus on partial geometric designs with block sizes three and four. First, we show that a partial geometric design with block size three is a 2‐design, or a transversal design, or each line of a generalized quadrangle that is repeated λ times. Second, we introduce the concept of type of partial geometric designs with block size four and characterize partial geometric designs of type [λ,1,1] for some integer λ≥1. [ABSTRACT FROM AUTHOR]

A pentagonal geometry PENT(k,r) is a partial linear space, where every line, or block, is incident with k points, every point is incident with r lines, and for each point x, there is a line incident with precisely those points that are not collinear with x. An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear with precisely those points on the other line. We give a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs. Also we present 39 new pentagonal geometries with block size 4 and five with block size 5, all with connected deficiency graphs. Consequentially we determine the existence spectrum up to a few possible exceptions for PENT(4,r) that do not contain opposite line pairs and for PENT(4,r) with one opposite line pair. More generally, given j we show that there exists a PENT(4,r) with j opposite line pairs for all sufficiently large admissible r. Using some new group divisible designs with block size 5 (including types 235,271, and 1023) we significantly extend the known existence spectrum for PENT(5,r). [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]

We complete the existence spectrum of perfect Mendelsohn designs PMD(v,5) as v≡0,1(mod5),v≠6,10 by exhibiting previously unknown designs PMD(15,5) and PMD(20,5). [ABSTRACT FROM AUTHOR]

Several varieties of quasigroups obtained from perfect Mendelsohn designs with block size 4 are defined. One of these is obtained from the so‐called directed standard construction and satisfies the law xy⋅(y⋅xy)=x and another satisfies Stein's third law xy⋅yx=y. Such quasigroups which satisfy the flexible law x⋅yx=xy⋅x are investigated and characterized. Quasigroups which satisfy both of the laws xy⋅(y⋅xy)=x and xy⋅yx=y are shown to exist. Enumeration results for perfect Mendelsohn designs PMD(9, 4) and PMD(12, 4) as well as for (nonperfect) Mendelsohn designs MD(8, 4) are given. [ABSTRACT FROM AUTHOR]

In this paper we relate t‐designs to a forbidden configuration problem in extremal set theory. Let 1t0ℓ denote a column of t 1's on top of ℓ 0's. Let q⋅1t0ℓ denote the (t+ℓ)×q matrix consisting of t rows of q 1's and ℓ rows of q 0's. We consider extremal problems for matrices avoiding certain submatrices. Let A be a (0, 1)‐matrix forbidding any (t+ℓ)×(λ+2) submatrix (λ+2)⋅1t0ℓ. Assume A is m‐rowed and only columns of sum t+1,t+2,...,m−ℓ are allowed to be repeated. Assume that A has the maximum number of columns subject to the given restrictions. Assume m is sufficiently large. Then A has each column of sum 0,1,...,t and m−ℓ+1,m−ℓ+2,...,m exactly once and, given the appropriate divisibility condition, the columns of sum t+1 correspond to a t‐design with block size t+1 and parameter λ. The proof derives a basic upper bound on the number of columns of A by a pigeonhole argument and then a careful argument, for large m, reduces the bound by a substantial amount down to the value given by design‐based constructions. We extend in a few directions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of large sets of symmetric partitioned incomplete latin squares of type gu (LSSPILSs) which can be viewed as a generalization of the well‐known golf designs. Constructions for LSSPILSs are presented from some other large sets, such as golf designs, large sets of group divisible designs, and large sets of Room frames. We prove that there exists an LSSPILS(gu) if and only if u ≥ 3, g(u − 1) ≡ 0 (mod 2), and (g, u) ≠ (1, 5). [ABSTRACT FROM AUTHOR]

A Ryser design has equally many points as blocks with the provision that every two blocks intersect in a fixed number of points λ. An improper Ryser design has only one replication number and is thus symmetric design. A proper Ryser design has two replication numbers. The only known construction of a Ryser design is the complementation of a symmetric design. Such a Ryser design is called a Ryser design of type 1. Let D denote a Ryser design of order v, index λ and replication numbers r1,r2. Let ei denote the number of points of D with replication number ri (with i=1,2). Call a block A small (respectively large) if |A|<2λ (respectively |A|>2λ) and average if |A|=2λ. Let D denote the integer e1−r2 and let ρ>1 denote the rational number r1−1r2−1. Main results of the present article are the following. For every block A, r1≥|A|≥r2 (this improves an earlier known inequality |A|≥r2). If there is no small block (respectively no large block) in D, then D≤−1 (respectively D≥0). With an extra assumption e2>e1 an earlier known upper bound on v is improved from a cubic to a quadratic in λ. It is also proved that if v≤λ2+λ+1 and if ρ equals λ or λ−1, then D is of type 1. Finally, a Ryser design with 2n+1 points is shown to be of type 1. [ABSTRACT FROM AUTHOR]

Hyperovals in projective planes turn out to have a link with t‐designs. Motivated by an unpublished work of Lonz and Vanstone, we present a construction for t‐designs and s‐resolvable t‐designs from hyperovals in projective planes of order 2n. We prove that the construction works for t≤5. In particular, for t=5 the construction yields a family of 5‐(2n+2,8,70(2n−2−1)) designs. For t=4 numerous infinite families of 4‐designs on 2n+2 points with block size 2k can be constructed for any k≥4. The construction assumes the existence of a 4‐(2n−1+1,k,λ) design, called the indexing design, including the complete 4‐(2n−1+1,k,(2n−1−3k−4)) design. Moreover, we prove that if the indexing design is s‐resolvable, then so is the constructed design. As a result, many of the constructed designs are s‐resolvable for s=2,3. We include a short discussion on the simplicity or non‐simplicity of the designs from hyperovals. [ABSTRACT FROM AUTHOR]

We determine a necessary and sufficient condition for the existence of semicyclic holey group divisible designs with block size three and group type (n,mt). New recursive constructions on semicyclic incomplete holey group divisible designs are introduced to settle this problem completely. [ABSTRACT FROM AUTHOR]

Plateaued functions on finite fields have been studied in many papers in recent years. As a generalization of plateaued functions on finite fields, we introduce the notion of a plateaued function on a finite abelian group. We will give a characterization of a plateaued function in terms of an equation of the matrix associated to the function. Then we establish a one‐to‐one correspondence between the Z2‐valued plateaued functions and partial geometric difference sets (with specific parameters) in finite abelian groups. We will also discuss two general methods (extension and lifting) for the construction of new partial geometric difference sets from old ones in (abelian or nonabelian) finite groups, and construct many partial geometric difference sets and plateaued functions. A one‐to‐one correspondence between partial geometric difference sets (in arbitrary finite groups) and partial geometric designs will be proved. [ABSTRACT FROM AUTHOR]

In this article, we mainly consider the existence problem of a group divisible design GDD(3,4,n+s) of type 1ns1. We present two recursive constructions for this configuration using candelabra systems and construct explicitly a few small examples admitting given automorphism groups. As an application, several new infinite classes of GDD(3,4,n+s)s of type 1ns1 are produced. Meanwhile a few new infinite families on candelabra quadruple systems with group sizes being odd and stem size greater than one are also obtained. [ABSTRACT FROM AUTHOR]

We deal with group divisible designs (GDDs) that have block size four and group type gum1, where g≡2 or 4 (mod 6). We show that the necessary conditions for the existence of a 4‐GDD of type gum1 are sufficient when g = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for g=2tqs, q = 19, 23, 25, 29, 31, s, t=1,2,..., as well as for g=2tq, q = 2, 5, 7, 11, 13, 17, t=1,2,..., with possible exceptions 569m1, 809m1 and 1129m1 for a few large values of m. [ABSTRACT FROM AUTHOR]

The crossing numberCR(G) of a graph G=(V,E) is the smallest number of edge crossings over all drawings of G in the plane. For any k≥1, the k‐planar crossing number of G,CRk(G), is defined as the minimum of CR(G1)+CR(G2)+⋯+CR(Gk) over all graphs G1,G2,...,Gk with ∪i=1kGi=G. Pach et al [Comput. Geom.: Theory Appl. 68 (2018), pp. 2–6] showed that for every k≥1, we have CRk(G)≤(2/k2−1/k3)CR(G) and that this bound does not remain true if we replace the constant 2/k2−1/k3 by any number smaller than 1/k2. We improve the upper bound to (1/k2)(1+o(1)) as k→∞. For the class of bipartite graphs, we show that the best constant is exactly 1/k2 for every k. The results extend to the rectilinear variant of the k‐planar crossing number. [ABSTRACT FROM AUTHOR]

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns B(n,t) that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows (n) and a maximum in absolute value correlation between any two columns (t∕n). In particular, they proved that B(n,2)≤n+2 for n≡2 (mod 4) and n>6. However, the only known SSD satisfying this upper bound is when n=10. By utilizing a computer search, we prove that B(n,2)≤n+1 for n=18,22,30, and B(14,2)=15. These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the E(s2) lower bound for SSDs with n rows and n+2 columns, for n=14,18,22, and 30. Finally, we show that a skew‐type Hadamard matrix of order n can be used to construct an SSD with n−2 rows and n−1 columns that proves B(n−2,2)≥n−1. Hence, we establish B(n,2)=n+1 for n=14,18,22,30 and B(n,2)≥n+1 for all n≡2 (mod 4) such that n≤270. Our result also implies that B(n,2)≥n+1 when n+1 is a prime power and n+1≡3 (mod 4). We conjecture that n+1=B(n,2)10 and n≡2 (mod 4), where B′(n,2) is the maximum number of equiangular lines in Rn−1 with pairwise angle arccos(2∕n). [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]

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable (v,4,6)‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable (v,4,3)‐BIBDs are v≡1 (mod 4) and v≥17. In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable (v,4,3)‐RBIBD, namely v≡1 (mod 4) and v≥9, are sufficient. A few new pairwise balanced designs are also given. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the notion of quasi‐orthogonal cocycle. This is motivated in part by the maximal determinant problem for square { ± 1 }‐matrices of size congruent to 2 modulo 4. Quasi‐orthogonal cocycles are analogous to the orthogonal cocycles of algebraic design theory. Equivalences with new and known combinatorial objects afforded by this analogy, such as quasi‐Hadamard groups, relative quasi‐difference sets, and certain partially balanced incomplete block designs, are proved. [ABSTRACT FROM AUTHOR]

Abstract: We derive a previously unknown lower bound of 41 for the frequency of s max = 8 of an E(s2)‐optimal and minimax‐optimal supersaturated design (SSD) with 20 rows and 76 columns. This is accomplished by an exhaustive computer search that uses the combinatorial properties of resolvable 2 − (20, 10, 36) designs and the parallel class intersection pattern method. We also classify all nonisomorphic E(s2)‐optimal 4‐circulant SSDs with 20 rows and s max = 8. [ABSTRACT FROM AUTHOR]

Abstract: We establish that the logarithm of the number of latin d‐cubes of order n is Θ ( n d ln n ) and the logarithm of the number of sets of t ( t ≥ 2 is fixed) orthogonal latin squares of order n is Θ ( n 2 ln n ). Similar estimations are obtained for systems of mutually strongly orthogonal latin d‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order n such that the logarithm of its cardinality is Θ ( n 3 ln n ) as n → ∞ and n mod 6 = 2 or 4. [ABSTRACT FROM AUTHOR]

The research on directed PBDs is motivated by the construction of t-deletion/insertion-correcting codes. Fuji-Hara, Miao, Wang, and Yin have determined the existence of directed PBDs with block sizes from the set [ABSTRACT FROM AUTHOR]

This paper addresses the question of whether triple arrays can be constructed from Youden squares developed from difference sets.We prove that if the difference set is abelian, then having -1 as multiplier is both a necessary and sufficient condition for the construction to work. Using this, we are able to give a new infinite family of triple arrays. We use an alternative version of the construction to analyze the case of non-abelian difference sets, for which we prove a sufficient condition for giving triple arrays. We do a computer search for such non-abelian difference sets, but have not found any examples satisfying the given condition. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to classify all pairs (D, G), where D is a nontrivial 2-(v, k, 2) design, and G ≤ Aut(D) acts transitively on the set of blocks of D and primitively on the set of points of D with sporadic socle. We prove that there exists only one such pair (D, G): D is the unique 2-(176,8,2) design and G = HS, the Higman-Sims simple group. [ABSTRACT FROM AUTHOR]

Multi-layered Youden rectangles are introduced. These new designs include double and triple Youden rectangles as subdesigns. Though many double Youden rectangles are known, the triples, introduced in 1994, have to now yielded few nontrivial examples. Two infinite series of multi-layered Youden rectangles are constructed, and so also many new triple Youden rectangles. [ABSTRACT FROM AUTHOR]

In this paper, we determine the necessary and sufficient conditions for the existence of an equitably ℓ-colorable balanced incomplete block design for any positive integer [ABSTRACT FROM AUTHOR]

We consider the existence problem for a semi-cyclic holey group divisible design of type [ABSTRACT FROM AUTHOR]

A cube design of order v, or a CUBE( v), is a decomposition of all cyclicly oriented quadruples of a v-set into oriented cubes. A CUBE( v) design is unoriented if its cubes can be paired so that the cubes in each pair are related by reflection through the center. A cube design is degenerate if it has repeated points on one of its cubes, otherwise it is nondegenerate. We show that a nondegenerate CUBE( v) design exists for all integers [ABSTRACT FROM AUTHOR]

Intersection numbers for subspace designs are introduced and q-analogs of the Mendelsohn and Köhler equations are given. As an application, we are able to determine the intersection structure of a putative q-analog of the Fano plane for any prime power q. It is shown that its existence implies the existence of a 2- [ABSTRACT FROM AUTHOR]

A [ABSTRACT FROM AUTHOR]

In recent years, several methods have been proposed for constructing [ABSTRACT FROM AUTHOR]

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k-pc and for a given k the number of k-pcs is denoted rk. In this paper, we consider the case of block sizes 3 and 4 (both existent). We use v to denote the number of points, in this case the necessary conditions imply that v ≡ 0 (mod 12). We prove that all admissible URDs with v < 200 points exist, with the possible exceptions of 13 values of r4 over all permissible v. We obtain a URD({3, 4}; 276) with r4 = 9 by direct construction use it to and complete the construction of all URD({3, 4}; v) with r4 = 9. We prove that all admissible URDs for v ≡ 36 (mod 144), v ≡ 0 (mod 60), v ≡ 36 (mod 108), and v ≡ 24 (mod 48) exist, with a few possible exceptions. Recently, the existence of URDs for all admissible parameter sets with v ≡ 0 (mod 48) was settled, this together with the latter result gives the existence all admissible URDs for v ≡ 0 (mod 24), with a few possible exceptions. [ABSTRACT FROM AUTHOR]

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

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

A cycle C in a graph G is extendable if there is some other cycle in G that contains each vertex of C plus one additional vertex. A graph is cycle extendable if every non-Hamilton cycle in the graph is extendable. A balanced incomplete block design, BIBD [ABSTRACT FROM AUTHOR]

It is well known that when [ABSTRACT FROM AUTHOR]

We show that, when the number of elements is a prime power q, in many situations the necessary conditions [ABSTRACT FROM AUTHOR]

This note provides a correction and some additions to a 1999 article by Luigia Berardi and Fulvio Zuanni on blocking 3-sets in designs. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 328-331, 2012 [ABSTRACT FROM AUTHOR]

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

The article presents the table of content for the issue of the periodical, along with information on commercial reprints and copyright and photocopying of the periodical.

In this note, we present a corrected proof of Lemma 5.8 which was wrong due to the incorrectness of Construction 2.3, from our paper. [ABSTRACT FROM AUTHOR]