An r‐golf design of order v , briefly by r‐G(v), is a large set of idempotent Latin squares of order v (ILS ( v ) s) which contains r symmetric ILS ( v ) s and v − r − 2 2 transposed pairs of ILS ( v ) s. In this paper, we mainly consider the existence problem of r‐G(v)s. We present several recursive constructions and also display some direct constructions. As an application, several infinite classes

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

A technique for classifying Butson‐type Hadamard matrices over the complex 4th roots of unity is developed. The technique resembles the approach used by Kharaghani and Tayfeh‐Rezaie in their classification of the (real) 32 × 32 Hadamard matrices, where the search is split into parts according to matrix types. A classification of the 18 × 18 Butson‐type Hadamard matrices over the complex 4th roots of

Let PG ( F q v ) be the ( v − 1 ) ‐dimensional projective space over F q and let Γ be a simple graph of order q k − 1 q − 1 for some k . A 2 − ( v , Γ , λ ) design over F q is a collection ℬ of graphs (blocks) isomorphic to Γ with the following properties: the vertex set of every block is a subspace of PG ( F q v ) ; every two distinct points of PG ( F q v ) are adjacent in exactly λ blocks. This new

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

Erdős and Lovász noticed that an r ‐uniform intersecting hypergraph H with maximal covering number, that is, τ ( H ) = r , must have at least 8 3 r − 3 edges. There has been no improvement on this lower bound for 45 years. We try to understand the reason by studying some small cases to see whether the truth lies very close to this simple bound. Let q ( r ) denote the minimum number of edges in an intersecting

Using group rings and characters as in the theory of abelian difference sets, some nonexistence results for strong difference families are provided. Existences of strong difference families with base block size 3 ≤ k ≤ 9 are discussed. Via strong difference families, eight 2‐ ( v , k , λ ) designs whose existences were unknown are constructed for ( v , k , λ ) ∈ { ( 3417 , 8 , 1 ) , ( 3753 , 8 , 1

We construct some large families of permutations of Z n such that the sum of any two permutations from a family is again a permutation of Z n (not necessarily in the same family). Our families are significantly larger than the largest such families known so far in the literature (depending on the value of n , the improvement can be as large as exponential). We also show that our families are maximal

A ( k , l ) ‐configuration is a set of l blocks on k points. For Steiner triple systems, ( l + 2 , l ) ‐configurations are of particular interest. The smallest nontrivial such configuration is the Pasch configuration, which is a ( 6 , 4 ) ‐configuration. A Steiner triple system of order v , an STS ( v ) , is r ‐sparse if it does not contain any ( l + 2 , l ) ‐configuration for 4 ≤ l ≤ r . The existence

It is shown that for v ≠ 7 , 8 , 11 , 12 , or 13, there exists an optimal covering with triples on v points that contains no Pasch configurations.

Let K n + I denote the complete graph of even order with a 1‐factor duplicated. The spouse‐loving variant of the Oberwolfach Problem, denoted O P + ( m 1 , m 2 , … , m t ) , asks for the existence of a 2‐factorization of K n + I in which each 2‐factor consists of cycles of length m i , for all i , 1 ≤ i ≤ t , such that n = m 1 + m 2 + ⋯ + m t . If m 1 = m 2 = ⋯ = m t = m , then the problem is denoted

A k‐cycle with a pendant edge attached to each vertex is called a k‐sun. The existence problem for k‐sun decompositions of Kv, with k odd, has been solved only when k = 3 or 5. By adapting a method used by Hoffmann, Lindner, and Rodger to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a k ‐sun system of K v ( k odd) whenever v lies in the range 2 k

A partial difference set S in a finite group G satisfying 1 ∉ S and S = S − 1 corresponds to an undirected strongly regular Cayley graph Cay ( G , S ) . While the case when G is abelian has been thoroughly studied, there are comparatively few results when G is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups

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 λ , denoted by TQS λ ( n ) , is a pair ( X , ℬ ) , where X is an n ‐set and ℬ is a set of oriented tetrahedra (blocks) such that every cyclic triple on X is contained

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

Many deep connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called k ‐nets (and in particular, between collections of MUBs and finite affine planes). Here we introduce the notion of a k ‐net over a C * ‐algebra, providing a common framework for both objects. In the commutative case, we recover (classical) k ‐nets, while the choice of M d

Let r and r′, (r > r′) be the two replication numbers of a λ ‐design D . We show that if r − r ′ is a prime power or 33, then D is a design of type‐1. We derive two inequalities, where equality holds if and only if D is a design of type‐1. Let D be a λ ‐design with two block sizes with v = n p + 1 points, where p is prime and 2 ⩽ n ⩽ 22 . We develop a procedure to obtain exceptions for such type‐1

In this paper, we use constructions of Heffter arrays to verify the existence of face 2‐colorable embeddings of cycle decompositions of the complete graph. Specifically, for n ≡ 1 ( mod 4 ) and k ≡ 3 ( mod 4 ) , n ≫ k ⩾ 7 and when n ≡ 0 ( mod 3 ) then k ≡ 7 ( mod 12 ) , there exist face 2‐colorable embeddings of the complete graph K 2 n k + 1 onto an orientable surface where each face is a cycle of

By Raaphorst et al, for a prime power q , covering arrays (CAs) with strength 3 and index 1, defined over the alphabet F q , were constructed using the output of linear feedback shift registers defined by cubic primitive polynomials in F q [ x ] . These arrays have 2 q 3 − 1 rows and q 2 + q + 1 columns. We generalize this construction to apply to all polynomials; provide a new proof that CAs are indeed

This paper introduces almost partitionable sets (APSs) to generalize the known concept of partitionable sets. These notions provide a unified frame to construct Z ‐cyclic patterned starter whist tournaments and cyclic balanced sampling plans excluding contiguous units. The existences of partitionable sets and APSs are investigated. As an application, a large number of optical orthogonal codes achieving

We investigate how Legendre G ‐array pairs are related to several different perfect binary G ‐array families. In particular we study the relations between Legendre G ‐array pairs, Sidelnikov‐Lempel‐Cohn‐Eastman Z q − 1 ‐arrays, Yamada‐Pott G ‐array pairs, Ding‐Helleseth‐Martinsen Z 2 × Z p m ‐arrays, Yamada Z ( q − 1 ) ∕ 2 ‐arrays, Szekeres Z p m ‐array pairs, Paley Z p m ‐array pairs, and Baumert

It is well known to us that a graph of diameter l has at least l + 1 eigenvalues. A graph is said to be Laplacian (resp, normalized Laplacian) l ‐extremal if it is of diameter l having exactly l + 1 distinct Laplacian (resp, normalized Laplacian) eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Each split graph is of diameter at most 3. In this paper

We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to be affine, nonramified. By exhibiting a one‐to‐one correspondence between isomorphism classes of affine MTS and those of modules over the Eisenstein integers, we solve the isomorphism problem for affine, nonramified MTS and enumerate these isomorphism classes (extending the

Let C 4 be a cycle of order 4. Write e x ( n , n , n , C 4 ) for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has n vertices that have no subgraph isomorphic to C 4 . In this paper, we show that e x ( n , n , n , C 4 ) ≥ 3 2 n ( p + 1 ) , where n = p ( p − 1 ) 2 and p is a prime number. Note that e x ( n , n , n , C 4 ) ≤ ( 3 2 2 + o ( 1 )

The dual codes of the ternary linear codes of the residual designs of biplanes on 56 points are used to prove the nonexistence of quasisymmetric 2‐ ( 56 , 12 , 9 ) and 2‐ ( 57 , 12 , 11 ) designs with intersection numbers 0 and 3, and the nonexistence of a 2‐ ( 267 , 57 , 12 ) quasi‐3 design. The nonexistence of a 2‐ ( 149 , 37 , 9 ) quasi‐3 design is also proved.

In this article, we study symmetric ( v , k , λ ) designs admitting a flag‐transitive and point‐primitive automorphism group G whose socle is isomorphic to a projective special linear group of dimension at most four. We, in particular, determine all such possible parameters ( v , k , λ ) and show that such a design belongs to one of two infinite families of point‐hyperplane designs or it is isomorphic

The maximum independence number of Steiner triple systems of order v is well‐known. Motivated by questions of access balancing in storage systems, we determine the maximum total cardinality of a pair of disjoint independent sets of Steiner triple systems of order v for all admissible orders.

Let X be a finite set with v elements, called points and β be a family of subsets of X , called blocks. A pair ( X , β ) is called λ ‐design whenever ∣ β ∣ = ∣ X ∣ and 1. for all B i , B j ∈ β , i ≠ j , ∣ B i ∩ B j ∣ = λ ; 2. for all B j ∈ β , ∣ B j ∣ = k j > λ , and not all k j are equal. The only known examples of λ ‐designs are so‐called type‐1 designs, which are obtained from symmetric designs

Group divisible designs (GDDs) with block size 4 and at most 30 points are known for all feasible group types except three, namely 2 3 5 4 , 3 5 6 2 , and 2 2 5 5 . In this paper we provide solutions for the first two of these three 4‐GDDs without assuming any automorphisms. We also construct several other 4‐GDDs. These include classes of 4‐GDDs of types ( 3 m ) 4 ( 6 m ) q ( 3 n ) 1 for 0 ≤ n ≤ (

Deep learning is a machine learning methodology using a multilayer neural network. Let V 1 , V 2 , … , V L be mutually disjoint node sets (layers). A multilayer neural network can be regarded as a union of the complete bipartite graphs K ∣ V i ∣ , ∣ V i + 1 ∣ on consecutive two node sets V i and V i + 1 for i = 1 , 2 , … , L − 1 . The edges of a bipartite graph function as weights which are represented

A Deza graph with parameters ( v , k , b , a ) is a k ‐regular graph on v vertices in which the number of common neighbors of two distinct vertices takes one of the following values: b or a , where b ≥ a . In the previous papers Deza graphs with b = k − 1 were characterized. In this paper, we characterize Deza graphs with b = k − 2 .

For positive integers n ≥ k ≥ t , a collection B of k ‐subsets of an n ‐set X is called a t ‐packing if every t ‐subset of X appears in at most one set in B . In this paper, we investigate the existence of the maximum 3‐packings whenever n is sufficiently larger than k . When n ≢ 2 ( mod k − 2 ) , the optimal value for the size of a 3‐packing is settled. In other cases, lower and upper bounds are obtained

A pair of orthogonal latin cubes of order q is equivalent to a maximum distance separable code with distance 3 or to an OA 1 ( 3 , 5 , q ) orthogonal array. We construct pairs of orthogonal latin cubes for sequences of previously unknown orders q i = 16 ( 18 i − 1 ) + 4 and q i ′ = 16 ( 18 i + 5 ) + 4 . The minimal new obtained parameters of orthogonal arrays are OA 1 ( 3 , 5 , 84 ) .

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 x y ⋅ ( y ⋅ x y ) = x and another satisfies Stein's third law x y ⋅ y x = y . Such quasigroups which satisfy the flexible law x ⋅ y x = x y ⋅ x are investigated and characterized. Quasigroups which satisfy

An e ‐star is a complete bipartite graph K 1 , e . An e ‐star system of order n > 1 , S e ( n ) , is a partition of the edges of the complete graph K n into e ‐stars. An e ‐star system is said to be k ‐colourable if its vertex set can be partitioned into k sets (called colour classes) such that no e ‐star is monochromatic. The system S e ( n ) is k ‐chromatic if S e ( n ) is k ‐colourable but is not

A partial Steiner triple system of order u is a pair ( U , A ) , where U is a set of u elements and A is a set of triples of elements of U such that any two elements of U occur together in at most one triple. If each pair of elements occur together in exactly one triple it is a Steiner triple system. An embedding of a partial Steiner triple system ( U , A ) is a (complete) Steiner triple system ( V

Let D ( n ) be the number of pairwise disjoint Steiner quadruple systems (SQS) of order n . A simple counting argument shows that D ( n ) ≤ n − 3 and a set of n − 3 such systems is called a large set. No nontrivial large set was constructed yet, although it is known that they exist if n ≡ 2 or 4 ( mod 6 ) is large enough. When n ≥ 7 and n ≡ 1 or 5 ( mod 6 ) , we present a recursive construction and

We show that the necessary conditions are sufficient for the existence of two disjoint near (hooked) Rosa sequences, with all admissible orders n ≥ 6 and all possible defects. Further, we apply this result for the existence of new types of cyclic and simple GDDs.

An n ‐ary k ‐radius sequence is a finite sequence of elements taken from an alphabet of size n in which any two distinct elements occur within distance k of each other somewhere in the sequence. The study of constructing short k ‐radius sequences was motivated by some problems occurring in large data transfer. Let f k ( n ) be the shortest length of any n ‐ary k ‐radius sequence. We show that the conjecture

In this paper we relate t ‐designs to a forbidden configuration problem in extremal set theory. Let 1 t 0 ℓ denote a column of t 1's on top of ℓ 0's. Let q ⋅ 1 t 0 ℓ 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

It is known that in any r‐coloring of the edges of a complete r‐uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on n vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3‐coloring of the edges? Gyárfás proved that ( 2 n + 3 ) / 3 is an absolute lower bound and that this lower bound is best possible for infinitely

Using a backtracking algorithm along with an essential change to the rows of representatives of known 13 710 027 equivalence classes of Hadamard matrices of order 32, we make an exhaustive computer search feasible and show that there are exactly 6662 inequivalent skew‐Hadamard matrices of order 32. Two skew‐Hadamard matrices are considered SH‐equivalent if they are similar by a signed permutation matrix

Qk is the simple graph whose vertices are the k‐tuples with entries in {0, 1} and edges are the pairs of k‐tuples that differ in exactly one position. In this paper, we proved that there exists a Q5‐factorization of λKn if and only if (a) n ≡ 0(mod 32) if λ ≡ 0(mod 5) and (b) n ≡ 96(mod 160) if λ ≢ 0(mod 5).

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)

Let F be a 2‐regular graph of order v. The Oberwolfach problem, OP(F), asks for a 2‐factorization of the complete graph on v vertices in which each 2‐factor is isomorphic to F. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group G. We will also consider

In a latin square of order n, a near transversal is a collection of n −1 cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.

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

A partial Steiner triple system of order n is sequenceable if there is a sequence of length n of its distinct points such that no proper segment of the sequence is a union of point‐disjoint blocks. We prove that if a partial Steiner triple system has at most three point‐disjoint blocks, then it is sequenceable.

In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce

In 1977, D. Betten defined a geometric group to be a permutation group ( G , Ω ) such that G = Aut ( R ) for some hypergraph R on Ω . In this paper, we extend Betten's notion of a geometric group to what we call a geometric group of second order. By definition, this is a permutation group for which G = Aut ( R ) for some set R = { R 1 , R 2 , … , R d } of hypergraphs on Ω . Our main focus will be on

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 2 n . We prove that the construction works for t ≤ 5 . In particular, for t = 5 the construction yields a family of 5‐ ( 2 n + 2 , 8 , 70 ( 2 n − 2 − 1 ) ) designs

We show that there is no (95, 40, 12, 20) strongly regular graph and, consequently, there is no (96, 45, 24, 18) strongly regular graph, no nontrivial regular two‐graph on 96 vertices, and no partial geometry pg(4, 9, 2). The main idea of the result is based on the star complement technique and requires a moderate amount of computation.

Combinatorial t ‐designs have wide applications in coding theory, cryptography, communications, and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a t ‐design. In this paper, we first determine the weight distributions of a class of linear codes derived from the dual of some extended cyclic codes. We then obtain infinite families of 2‐designs

In this paper, we study the existence problem for cyclic ℓ ‐cycle decompositions of the graph K m [ n ] , the complete multipartite graph with m parts of size n , and give necessary and sufficient conditions for their existence in the case that 2 ℓ | ( m − 1 ) n .

In this paper, we define a new class of partially filled arrays, called relative Heffter arrays, that are a generalization of the Heffter arrays introduced by Archdeacon in 2015. Let v = 2 n k + t be a positive integer, where t divides 2 n k , and let J be the subgroup of Z v of order t . A H t ( m , n ; s , k ) Heffter array over Z v relative to J is an m × n partially filled array with elements in

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. However, determining the Lagrangian density of a hypergraph is not an easy task even for a “simple” hypergraph. For example, to determine the Lagrangian density of K 4 3 is equivalent to determine the Turán

For a family F of r ‐graphs, let ex ( n , F ) denote the maximum number of edges in an F ‐free r ‐graph on n vertices. Let F r ( v , e ) denote the family of all r ‐graphs with e edges and at most v vertices. We prove that ex ( n , F r ( r + 1 , 2 ) ∪ F r ( r + 2 , 3 ) ) = ( 1 r − o ( 1 ) ) ( n r − 1 ) .

Partial difference sets with parameters ( v , k , λ , μ ) = ( v , ( v − 1 ) / 2 , ( v − 5 ) / 4 , ( v − 1 ) / 4 ) are called Paley type partial difference sets. In this note, we prove that if there exists a Paley type partial difference set in an abelian group of order v, where v is not a prime power, then v = n 4 or 9 n 4 , n > 1 an odd integer. In 2010, Polhill constructed Paley type partial difference

In this note, an elementary proof of a generalized Drápal–Valent type‐inequality is presented.

We investigate the upper chromatic number of the hypergraph formed by the points and the k ‐dimensional subspaces of PG ( n , q ) ; that is, the most number of colors that can be used to color the points so that every k ‐subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct