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  • Partitionable sets, almost partitionable sets, and their applications
    J. Comb. Des. (IF 0.731) Pub Date : 2020-07-20
    Yanxun Chang; Simone Costa; Tao Feng; Xiaomiao Wang

    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

    更新日期:2020-07-21
  • Legendre G‐array pairs and the theoretical unification of several G‐array families
    J. Comb. Des. (IF 0.731) Pub Date : 2020-07-15
    K. T. Arasu; D. A. Bulutoglu; J. R. Hollon

    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

    更新日期:2020-07-16
  • On split graphs with three or four distinct (normalized) Laplacian eigenvalues
    J. Comb. Des. (IF 0.731) Pub Date : 2020-07-13
    Shuchao Li; Wanting Sun

    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

    更新日期:2020-07-14
  • Combinatorial designs for deep learning
    J. Comb. Des. (IF 0.731) Pub Date : 2020-05-04
    Shoko Chisaki; Ryoh Fuji‐Hara; Nobuko Miyamoto

    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

    更新日期:2020-07-07
  • Deza graphs with parameters (v,k,k−2,a)
    J. Comb. Des. (IF 0.731) Pub Date : 2020-05-03
    Vladislav V. Kabanov; Leonid Shalaginov

    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 .

    更新日期:2020-07-07
  • Exceptions and characterization results for type‐1 λ‐designs
    J. Comb. Des. (IF 0.731) Pub Date : 2020-05-18
    Ajeet Kumar Yadav; Rajendra M. Pawale; Mohan S. Shrikhande

    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

    更新日期:2020-07-07
  • Symmetric designs and projective special linear groups of dimension at most four
    J. Comb. Des. (IF 0.731) Pub Date : 2020-06-07
    Seyed Hassan Alavi; Mohsen Bayat; Asharf Daneshkhah

    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

    更新日期:2020-07-07
  • Affine Mendelsohn triple systems and the Eisenstein integers
    J. Comb. Des. (IF 0.731) Pub Date : 2020-06-24
    Alex W. Nowak

    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

    更新日期:2020-06-25
  • A note on 3‐partite graphs without 4‐cycles
    J. Comb. Des. (IF 0.731) Pub Date : 2020-06-22
    Zequn Lv; Mei Lu; Chunqiu Fang

    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 )

    更新日期:2020-06-23
  • Ternary codes, biplanes, and the nonexistence of some quasisymmetric and quasi‐3 designs
    J. Comb. Des. (IF 0.731) Pub Date : 2020-06-22
    Akihiro Munemasa; Vladimir D. Tonchev

    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.

    更新日期:2020-06-23
  • On the maximum double independence number of Steiner triple systems
    J. Comb. Des. (IF 0.731) Pub Date : 2020-05-29
    Dylan Lusi; Charles J. Colbourn

    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.

    更新日期:2020-05-29
  • Some new group divisible designs with block size 4 and two or three group sizes
    J. Comb. Des. (IF 0.731) Pub Date : 2020-05-14
    R. Julian R. Abel; Yudhistira A. Bunjamin; Diana Combe

    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 ≤ (

    更新日期:2020-05-14
  • Packing cliques in 3‐uniform hypergraphs
    J. Comb. Des. (IF 0.731) Pub Date : 2020-04-22
    Ramin Javadi; Ehsan Poorhadi; Farshad Fallah

    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

    更新日期:2020-04-22
  • Constructions of pairs of orthogonal latin cubes
    J. Comb. Des. (IF 0.731) Pub Date : 2020-04-16
    Vladimir N. Potapov

    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 ) .

    更新日期:2020-04-16
  • Quasigroups constructed from perfect Mendelsohn designs with block size 4
    J. Comb. Des. (IF 0.731) Pub Date : 2020-04-04
    Terry S. Griggs; Aleš Drápal; Andrew R. Kozlik

    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

    更新日期:2020-04-04
  • Colourings of star systems
    J. Comb. Des. (IF 0.731) Pub Date : 2020-03-28
    Iren Darijani; David A. Pike

    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

    更新日期:2020-03-28
  • On determining when small embeddings of partial Steiner triple systems exist
    J. Comb. Des. (IF 0.731) Pub Date : 2020-03-17
    Darryn Bryant; Ajani De Vas Gunasekara; Daniel Horsley

    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 (

    更新日期:2020-03-17
  • An improved recursive construction for disjoint Steiner quadruple systems
    J. Comb. Des. (IF 0.731) Pub Date : 2020-03-13
    Tuvi Etzion; Junling Zhou

    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

    更新日期:2020-03-13
  • The existence of disjoint (hooked) near‐Rosa sequences and applications
    J. Comb. Des. (IF 0.731) Pub Date : 2020-03-11
    Beth A. Austin; Nabil Shalaby

    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.

    更新日期:2020-03-11
  • Short k‐radius sequences, k‐difference sequences and universal cycles
    J. Comb. Des. (IF 0.731) Pub Date : 2020-03-09
    Xiande Zhang

    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

    更新日期:2020-03-09
  • Design theory and some forbidden configurations
    J. Comb. Des. (IF 0.731) Pub Date : 2020-03-06
    R. P. Anstee; Farzin Barekat; Zachary Pellegrin

    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

    更新日期:2020-03-06
  • Large monochromatic components in 3‐edge‐colored Steiner triple systems
    J. Comb. Des. (IF 0.731) Pub Date : 2020-02-27
    Louis DeBiasio; Michael Tait

    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

    更新日期:2020-02-27
  • Classification of skew‐Hadamard matrices of order 32 and association schemes of order 31
    J. Comb. Des. (IF 0.731) Pub Date : 2020-02-14
    Akihide Hanaki; Hadi Kharaghani; Ali Mohammadian; Behruz Tayfeh‐Rezaie

    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

    更新日期:2020-02-14
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