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     

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 .     

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. .

     

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.     

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).     

Enclosings of λ‐Fold 5‐Cycle Systems: Adding One Vertex

A k‐cycle system of a multigraph G is an ordered pair (V, C) where V is the vertex set of G and C is a set of k‐cycles, the edges of which partition the edges of G. A k‐cycle system of is known as a λ‐fold k‐cycle system of order V. A k‐cycle system of (V, C) is said to be enclosed in a k‐cycle system of if and . We settle the difficult enclosing problem for λ‐fold 5‐cycle systems with u = 1.     

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     

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     

A Doubling Construction for 3‐Uniform Friendship Hypergraphs with the Universal Pairs Property

In this note, we present a simple doubling construction for 3‐uniform friendship hypergraphs which generalizes the cubeconstructed hypergraphs from another study (L. Jørgensen and A. Sillasen, J Combin Designs (2014)). As a by‐product, we build point‐transitive 3‐uniform friendship hypergraphs of sizes and for all .     

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.