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 .     

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.     

The Existence and Construction of (K5∖e)‐Designs of Orders 27, 135, 162, and 216

The problem of the existence of a decomposition of the complete graph into disjoint copies of has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a ‐design. I show that divides 2^k3 for some and that . I construct ‐designs by prescribing as an automorphism group, and show that up to isomorphism there are exactly 24 ‐designs with as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed . Finally, the existence of ‐designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864.     

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.

     

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 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     

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. .

     

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.     

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.     

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     

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.     

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.     

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     

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.     

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.     

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 .     

Doubly Resolvable Nearly Kirkman Triple Systems

Necessary conditions for existence of a resolvable group divisible design (GDD) with block size 3 and type (a nearly Kirkman triple system, NKTS(v)), are and (mod 6). In this paper, we look at doubly resolvable NKTS(v)s; here we find that these necessary conditions are sufficient, except possibly for 64 values of v.     

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.     

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.