Group Divisible Designs with Block Size Four and Group Type for

Nonuniform group divisible designs (GDDs) have been studied by numerous researchers for the past two decades due to their essential role in the constructions for other types of designs. In this paper, we investigate the existence problem of ‐GDDs of type for . First, we determine completely the spectrum of ‐GDDs of types and . Furthermore, for general cases, we show that for each and , a ‐GDD of type exists if and only if , and , except possibly for , and .     

Detachments of Amalgamated 3‐Uniform Hypergraphs Factorization Consequences

A is a hypergraph obtained from by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph can be thought of as taking , partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph . In this paper, we use Nash‐Williams lemma on laminar families to prove a detachment theorem for amalgamated 3‐uniform hypergraphs, which yields a substantial generalization of previous amalgamation theorems by Hilton, Rodger, and Nash‐Williams. To demonstrate the power of our detachment theorem, we show that the complete 3‐uniform n‐partite multihypergraph can be expressed as the union of k edge‐disjoint factors, where for , is ‐regular, if and only if:

1. for all ,
2. for each i, , and
3. .

     

Finiteness Results in Triangle‐Free Quasi‐Symmetric Designs

Triangle‐free quasi‐symmetric 2‐ designs with intersection numbers ; and are investigated. Possibility of triangle‐free quasi‐symmetric designs with or is ruled out. It is also shown that, for a fixed x and a fixed ratio , there are only finitely many triangle‐free quasi‐symmetric designs. © 2012 Wiley Periodicals, Inc. J Combin Designs 00: 1‐6, 2012     

Simple Signed Steiner Triple Systems

Let X be a v‐set, be a set of 3‐subsets (triples) of X, and be a partition of with . The pair is called a simple signed Steiner triple system, denoted by ST, if the number of occurrences of every 2‐subset of X in triples is one more than the number of occurrences in triples . In this paper, we prove that exists if and only if , , and , where and for , . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012     

Semiframes with Block Size Three and Odd Group Size

A ‐semiframe of type is a ‐GDD of type , , in which the collection of blocks can be written as a disjoint union where is partitioned into parallel classes of and is partitioned into holey parallel classes, each holey parallel class being a partition of for some . A ‐SF is a ‐semiframe of type in which there are p parallel classes in and d holey parallel classes with respect to . In this paper, we shall show that there exists a (3, 1)‐SF for any if and only if , , , and .     

On the Enumeration of ‐Optimal and Minimax‐Optimal k‐Circulant Supersaturated Designs

In recent years, several methods have been proposed for constructing ‐optimal and minimax‐optimal supersaturated designs (SSDs). However, until now the enumeration problem of such designs has not been yet considered. In this paper, ‐optimal and minimax‐optimal k‐circulant SSDs with 6, 10, 14, 18, 22, and 26 runs, factors and are enumerated in a computer search. We have also enumerated all ‐optimal and minimax‐optimal k‐circulant SSDs with (mod 4) and . The computer search utilizes the fact that theses designs are equivalent to certain 1‐rotational resolvable balanced incomplete block designs. Combinatorial properties of these resolvable designs are used to restrict the search space.     

Some New Results on ‐Cyclic Patterned Starter Whist Tournaments and Related Frames

The whist tournament problem was introduced by Moore in 1896. To construct ‐cyclic whist tournaments, ‐cyclic patterned starter whist tournaments were introduced. The systematic research on ‐cyclic patterned starter whist tournaments dates back to 1954. Recently, ‐cyclic patterned starter whist tournament frames were shown to be instrumental in the constructions of ‐cyclic patterned starter whist tournaments. In this paper, we develop some necessary conditions for the existence of ‐cyclic patterned starter whist tournament frames and some new necessary conditions for the existence of ‐cyclic patterned starter whist tournaments. An infinite family of ‐cyclic patterned starter whist tournament frames is constructed through Weil's theorem on character sums. As a consequence, a new infinite family of ‐cyclic patterned starter whist tournaments is obtained. In addition, we also give a near complete solution to the existence of ‐cyclic patterned starter whist tournaments on no greater than 300 players.     

Partitions of V(n,q) into 2‐ and s‐Dimensional Subspaces

Let denote a vector space of dimension n over the field with q elements. A set of subspaces of V is a (vector space) partition of V if every nonzero element of V is contained in exactly one subspace in . Suppose that is a partition of V with subspaces of dimension for . Then we call the type of the partition . Which possible types correspond to actual partitions is in general an open question. We prove that for any odd integer and for any integer , the existence of partitions of across a suitable range of types guarantees the existence of partitions of of essentially all the types for any integer . We then apply this result to construct new classes of partitions of V. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 467‐482, 2012     

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a ‐design of order n. The existence problem of ‐designs has been completely solved for the graphs for , for , K_{2, 3} and K_{3, 3}. In this paper, I prove that for all , if there exists a ‐design of order N, then there exists a ‐design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.     

Cycle Frames of Complete Multipartite Multigraphs ‐ III

For two graphs G and H their wreath product has vertex set in which two vertices and are adjacent whenever or and . Clearly, , where is an independent set on n vertices, is isomorphic to the complete m‐partite graph in which each partite set has exactly n vertices. A 2‐regular subgraph of the complete multipartite graph containing vertices of all but one partite set is called partial 2‐factor. For an integer λ, denotes a graph G with uniform edge multiplicity λ. Let J be a set of integers. If can be partitioned into edge‐disjoint partial 2‐factors consisting cycles of lengths from J, then we say that has a ‐cycle frame. In this paper, we show that for and , there exists a ‐cycle frame of if and only if and . In fact our results completely solve the existence of a ‐cycle frame of .     

Uniform Semi‐Latin Squares and Their Schur‐Optimality

Let n and k be integers, with and . An semi‐Latin square S is an array, whose entries are k‐subsets of an ‐set, the set of symbols of S, such that each symbol of S is in exactly one entry in each row and exactly one entry in each column of S. Semi‐Latin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. We say that an semi‐Latin square S is uniform if there is a constant μ such that any two entries of S, not in the same row or column, intersect in exactly μ symbols (in which case ). We prove that a uniform semi‐Latin square is Schur‐optimal in the class of semi‐Latin squares, and so is optimal (for use as an experimental design) with respect to a very wide range of statistical optimality criteria. We give a simple construction to make an semi‐Latin square S from a transitive permutation group G of degree n and order , and show how certain properties of S can be determined from permutation group properties of G. If G is 2‐transitive then S is uniform, and this provides us with Schur‐optimal semi‐Latin squares for many values of n and k for which optimal semi‐Latin squares were previously unknown for any optimality criterion. The existence of a uniform semi‐Latin square for all integers is shown to be equivalent to the existence of mutually orthogonal Latin squares (MOLS) of order n. Although there are not even two MOLS of order 6, we construct uniform, and hence Schur‐optimal, semi‐Latin squares for all integers . & 2012 Wiley Periodicals, Inc. J. Combin. Designs 00: 1–13, 2012     

Turyn‐Type Sequences: Classification,Enumeration, and Construction

Turyn‐type sequences, , are quadruples of ‐sequences , with lengths , respectively, where the sum of the nonperiodic autocorrelation functions of and twice that of is a δ‐function (i.e., vanishes everywhere except at 0). Turyn‐type sequences are known to exist for all even n not larger than 36. We introduce a definition of equivalence to construct a canonical form for in general. By using this canonical form, we enumerate the equivalence classes of for . We also construct the first example of Turyn‐type sequences .     

On the Choice Number of Packings

In this note, we show that for positive integers s and k, there is a function such that every t‐ packing with at least edges, , has choice number greater than s. Consequently, for integers s, k, t, and λ there is a such that every t‐ design with has choice number greater than s. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 504‐507, 2012     

Direct Constructions for General Families of Cyclic Mutually Nearly Orthogonal Latin Squares

Two Latin squares and , of even order n with entries , are said to be nearly orthogonal if the superimposition of L on M yields an array in which each ordered pair , and , occurs at least once and the ordered pair occurs exactly twice. In this paper, we present direct constructions for the existence of general families of three cyclic mutually orthogonal Latin squares of orders , , and . The techniques employed are based on the principle of Methods of Differences and so we also establish infinite classes of “quasi‐difference” sets for these orders.     

Flag‐Transitive 2‐ Symmetric Designs with Sporadic Socle

Let be a nontrivial 2‐ symmetric design admitting a flag‐transitive, point‐primitive automorphism group G of almost simple type with sporadic socle. We prove that there are up to isomorphism six designs, and must be one of the following: a 2‐(144, 66, 30) design with or , a 2‐(176, 50, 14) design with , a 2‐(176, 126, 90) design with or , or a 2‐(14,080, 12,636, 11,340) design with .     

On Quasi‐Hermitian Varieties

Quasi‐Hermitian varieties in are combinatorial generalizations of the (nondegenerate) Hermitian variety so that and have the same size and the same intersection numbers with hyperplanes. In this paper, we construct a new family of quasi‐Hermitian varieties. The isomorphism problem for the associated strongly regular graphs is discussed for .     

On Small Complete Arcs and Transitive ‐Invariant Arcs in the Projective Plane

Let q be an odd prime power such that q is a power of 5 or (mod 10). In this case, the projective plane admits a collineation group G isomorphic to the alternating group A₅. Transitive G‐invariant 30‐arcs are shown to exist for every . The completeness is also investigated, and complete 30‐arcs are found for . Surprisingly, they are the smallest known complete arcs in the planes , and . Moreover, computational results are presented for the cases and . New upper bounds on the size of the smallest complete arc are obtained for .     

Embedding Partial k‐Star Designs

A k‐star is the complete bipartite graph . Let G and H be graphs, and let be a partial H‐decomposition of G. A partial H‐decomposition, , of another graph is called an embedding of provided that and G is a subgraph of . We find an embedding of a partial k‐star decomposition of into a k‐star decomposition of , where s is at most if k is odd, and if k is even.     

The small Kakeya sets in,a conic

A Kakeya set in the linear representation , a nonsingular conic, is the point set covered by a set of lines, one through each point of . In this article, we classify the small Kakeya sets in . The smallest Kakeya sets have size , and all Kakeya sets with weight less than are classified: there are approximately types.     

Solution to An Isotopism Question Concerning Rank 2 Semifields

In [8] Dempwolff gives a construction of three classes of rank two semifields of order , with q and n odd, using Dembowski–Ostrom polynomials. The question whether these semifields are new, i.e. not isotopic to previous constructions, is left as an open problem. In this paper we solve this problem for , in particular we prove that two of these classes, labeled and , are new for , whereas presemifields in family are isotopic to Generalized Twisted Fields for each .