‐Relative Difference Sets and Their Representations

We show that every ‐relative difference set D in relative to can be represented by a polynomial , where is a permutation for each nonzero a. We call such an f a planar function on . The projective plane Π obtained from D in the way of M. J. Ganley and E. Spence (J Combin Theory Ser A, 19(2) (1975), 134–153) is coordinatized, and we obtain necessary and sufficient conditions of Π to be a presemifield plane. We also prove that a function f on with exactly two elements in its image set and is planar, if and only if, for any .     

Symmetric ‐Designs and ‐Difference Sets

In this paper, we introduce a method to construct ‐designs, which are also known as partial geometric designs, by using subsets of certain finite groups. We introduce the concept of ‐difference sets and investigate the existence and nonexistence of these structures. We also provide some nonexistence results on ‐designs based on the fact that ‐designs yield directed strongly regular graphs.     

The Mendelsohn Triple Systems of Order 13

A triple cyclically contains the ordered pairs , , , and no others. A Mendelsohn triple system of order v, or , is a set V together with a collection of ordered triples of distinct elements from V, such that and each ordered pair with is cyclically contained in exactly λ ordered triples. By means of a computer search, we classify all Mendelsohn triple systems of order 13 with ; there are 6 855 400 653 equivalence classes of such systems.     

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.     

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 .     

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.

     

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 .     

Neighborhoods in Maximum Packings of 2Kn and Quadratic Leaves of Triple Systems

In this paper, two related problems are completely solved, extending two classic results by Colbourn and Rosa. In any partial triple system of , the neighborhood of a vertex v is the subgraph induced by . For (mod 3) with , it is shown that for any 2‐factor F on or vertices, there exists a maximum packing of with triples such that F is the neighborhood of some vertex if and only if , thus extending the corresponding result for the case where or 1 (mod 3) by Colbourn and Rosa. This result, along with the companion result of Colbourn and Rosa, leads to a complete characterization of quadratic leaves of λ‐fold partial triple systems for all , thereby extending the solution where by Colbourn and Rosa.     

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.     

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.     

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.     

Constructions of Strictly m‐Cyclic and Semi‐Cyclic

An is a triple , where X is a set of points, is a partition of X into m disjoint sets of size n and is a set of 4‐element transverses of , such that each 3‐element transverse of is contained in exactly one of them. If the full automorphism group of an admits an automorphism α consisting of n cycles of length m (resp. m cycles of length n), then this is called m‐cyclic (resp. semi‐cyclic). Further, if all block‐orbits of an m‐cyclic (resp. semi‐cyclic) are full, then it is called strictly cyclic. In this paper, we construct some infinite classes of strictly m‐cyclic and semi‐cyclic , and use them to give new infinite classes of perfect two‐dimensional optical orthogonal codes with maximum collision parameter and AM‐OPPTS/AM‐OPPW property.     

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 .     

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).     

Some New Infinite Classes of Candelabra Quadruple Systems

Candelabra quadruple systems (CQS) were first introduced by Hanani who used them to determine the existence of Steiner quadruple systems. In this paper, a new method has been developed by constructing partial candelabra quadruple systems with odd group size, which is a generalization of the even cases, to complete a design. New results of candelabra quadruple systems have been obtained, i.e. we show that for any , there exists a CQS for all , and .     

Group Divisible Covering Designs with Block Size 4: A Type of Covering Array with Row Limit

A k‐GDCD, group divisible covering design, of type is a triple , where V is a set of gu elements, is a partition of V into u sets of size g, called groups, and is a collection of k‐subsets of V, called blocks, such that every pair of elements in V is either contained in a unique group or there is at least one block containing it, but not both. This family of combinatorial objects is equivalent to a special case of the graph covering problem and a generalization of covering arrays, which we call CARLs. In this paper, we show that there exists an integer such that for any positive integers g and , there exists a 4‐GDCD of type which in the worst case exceeds the Schönheim lower bound by δ blocks, except maybe when (1) and , or (2) , , and or . To show this, we develop constructions of 4‐GDCDs, which depend on two types of ingredients: essential, which are used multiple times, and auxiliary, which are used only once in the construction. If the essential ingredients meet the lower bound, the products of the construction differ from the lower bound by as many blocks as the optimal size of the auxiliary ingredient differs from the lower bound.     

Orthogonally Resolvable Cycle Decompositions

If a cycle decomposition of a graph G admits two resolutions, and , such that for each resolution class and , then the resolutions and are said to be orthogonal. In this paper, we introduce the notion of an orthogonally resolvable cycle decomposition, which is a cycle decomposition admitting a pair of orthogonal resolutions. An orthogonally resolvable cycle decomposition of a graph G may be represented by a square array in which each cell is either empty or filled with a k–cycle from G, such that every vertex appears exactly once in each row and column of the array and every edge of G appears in exactly one cycle. We focus mainly on orthogonal k‐cycle decompositions of and (the complete graph with the edges of a 1‐factor removed), denoted . We give general constructions for such decompositions, which we use to construct several infinite families. We find necessary and sufficient conditions for the existence of an OCD(n, 4). In addition, we consider orthogonal cycle decompositions of the lexicographic product of a complete graph or cycle with . Finally, we give some nonexistence results.     

Constructing Minimal Blocking Sets Using Field Reduction

We present a construction for minimal blocking sets with respect to ‐spaces in , the ‐dimensional projective space over the finite field of order . The construction relies on the use of blocking cones in the field reduced representation of , extending the well‐known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino, and Storme (the MPS‐construction); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in F. Mazzocca and O. Polverino, J Algebraic Combin 24(1) (2006), 61–81. Furthermore, we show that every minimal blocking set with respect to the hyperplanes in can be obtained by applying field reduction to a minimal blocking set with respect to ‐spaces in . We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS‐constructionthen it is of Rédei‐type, whereas a small minimal blocking set arises from our cone construction if and only if it is linear.     

Cycle Extensions in BIBD Block‐Intersection Graphs

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, consists of a set V of v elements and a block set of k‐subsets of V such that each 2‐subset of V occurs in exactly λ of the blocks of . The block‐intersection graph of a design is the graph having as its vertex set and such that two vertices of are adjacent if and only if their corresponding blocks have nonempty intersection. In this paper, we prove that the block‐intersection graph of any BIBD is cycle extendable. Furthermore, we present a polynomial time algorithm for constructing cycles of all possible lengths in a block‐intersection graph.     

Further Results on Large Sets of Resolvable Idempotent Latin Squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the off‐diagonal cells can be resolved into disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of RILS(v)s pairwise agreeing on only the main diagonal. In this paper, it is established that there exists an LRILS(v) for any positive integer , except for , and except possibly for .