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.     

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 .     

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     

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     

Some New Orders of Hadamard and Skew‐Hadamard Matrices

We construct Hadamard matrices of orders and , and skew‐Hadamard matrices of orders and . As far as we know, such matrices have not been constructed previously. The constructions use the Goethals–Seidel array, suitable supplementary difference sets on a cyclic group and a new efficient matching algorithm based on hashing techniques.     

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 .     

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 .     

Small Complete Caps from Singular Cubics

New families of complete caps in finite Galois spaces are obtained. For most pairs with and , they turn out to be the smallest known complete caps in . Our constructions rely on the bicovering properties of certain plane arcs contained in plane cubic curves with a cusp.     

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. .

     

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 .     

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 .     

Minimal Symmetric Differences of Lines in Projective Planes

Let q be an odd prime power and let be the minimum size of the symmetric difference of r lines in the Desarguesian projective plane . We prove some results about the function , in particular showing that there exists a constant such that for .     

Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs I: Latin Square Constructions

In an earlier paper the authors constructed a hamilton cycle embedding of in a nonorientable surface for all and then used these embeddings to determine the genus of some large families of graphs. In this two‐part series, we extend those results to orientable surfaces for all . In part I, we explore a connection between orthogonal latin squares and embeddings. A product construction is presented for building pairs of orthogonal latin squares such that one member of the pair has a certain hamiltonian property. These hamiltonian squares are then used to construct embeddings of the complete tripartite graph on an orientable surface such that the boundary of every face is a hamilton cycle. This construction works for all such that and for every prime p. Moreover, it is shown that the latin square construction utilized to get hamilton cycle embeddings of can also be used to obtain triangulations of . Part II of this series covers the case for every prime p and applies these embeddings to obtain some genus results.     

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

A pseudo‐hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo‐hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that is flag‐transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).     

Latin Squares with a Unique Intercalate

Suppose that and . We construct a Latin square of order n with the following properties:

• has no proper subsquares of order 3 or more .
• has exactly one intercalate (subsquare of order 2) .
• When the intercalate is replaced by the other possible subsquare on the same symbols, the resulting Latin square is in the same species as .

Hence generalizes the square that Sade famously found to complete Norton's enumeration of Latin squares of order 7. In particular, is what is known as a self‐switching Latin square and possesses a near‐autoparatopism.     

All λ‐Designs with Small λ are Type‐1

A λ‐design is a family of subsets of such that for all and not all are of the same size. Ryser's and Woodall's λ‐design conjecture states that each λ‐design can be obtained from a symmetric block design by a certain complementation procedure. Our main result is that the conjecture is true when λ < 63. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 408–431, 2012     

Decomposing Complete Equipartite Multigraphs into Cycles of Variable Lengths: The Amalgamation‐detachment Approach

Using the technique of amalgamation‐detachment, we show that the complete equipartite multigraph can be decomposed into cycles of lengths (plus a 1‐factor if the degree is odd) whenever there exists a decomposition of into cycles of lengths (plus a 1‐factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph .     

Set Systems with L‐Intersections and k‐Wise L‐Intersecting Families

In this paper, by employing linear algebra methods we obtain the following main results:

• (i) Let and be two disjoint subsets of such that Suppose that is a family of subsets of such that for every pair and for every i. Then Furthermore, we extend this theorem to k‐wise L‐intersecting and obtain the corresponding result on two cross L‐intersecting families. These results show that Snevily's conjectures proposed by Snevily (2003) are true under some restricted conditions. This result also gets an improvement of a theorem of Liu and Hwang (2013).
• (ii) Let p be a prime and let and be two subsets of such that or and Suppose that is a family of subsets of [n] such that (1) for every pair (2) for every i. Then This result improves the existing upper bound substantially.

     

Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of Ding‐Yuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.     

Three‐Phase Barker Arrays

A 3‐phase Barker array is a matrix of third roots of unity for which all out‐of‐phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two‐dimensional 3‐phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3‐phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double‐exponentially growing arithmetic function T such that no 3‐phase Barker array of size with exists for all . For example, , , and . When both dimensions are divisible by 3, the existence problem is settled completely: if a 3‐phase Barker array of size exists, then .