55 research outputs found
Resolvable Mendelsohn designs and finite Frobenius groups
We prove the existence and give constructions of a -fold perfect
resolvable -Mendelsohn design for any integers with such that there exists a finite Frobenius group whose kernel
has order and whose complement contains an element of order ,
where is the least prime factor of . Such a design admits as a group of automorphisms and is perfect when is a
prime. As an application we prove that for any integer in prime factorization, and any prime dividing
for , there exists a resolvable perfect -Mendelsohn design that admits a Frobenius group as a group of
automorphisms. We also prove that, if is even and divides for
, then there are at least resolvable -Mendelsohn designs that admit a Frobenius group as a group of
automorphisms, where is Euler's totient function.Comment: Final versio
Linear spaces with many small lines
AbstractIn this paper some of the work in linear spaces in which most of the lines have few points is surveyed. This includes existence results, blocking sets and embeddings. Also, it is shown that any linear space of order v can be embedded in a linear space of order about 13v in which there are no lines of size 2
Uniform hypergraphs containing no grids
A hypergraph is called an rĆr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Aiā©Aj=Biā©Bj=Ļ for 1ā¤i<jā¤r and {pipe}Aiā©Bj{pipe}=1 for 1ā¤i, jā¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1ā©C2{pipe}={pipe}C2ā©C3{pipe}={pipe}C3ā©C1{pipe}=1, C1ā©C2ā C1ā©C3. A hypergraph is linear, if {pipe}Eā©F{pipe}ā¤1 holds for every pair of edges Eā F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For rā„. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Ā© 2013 Elsevier Ltd
Resolvability of infinite designs
In this paper we examine the resolvability of infinite designs. We show that in stark contrast to the finite case, resolvability for infinite designs is fairly commonplace. We prove that every t-(v,k,Ī) design with t finite, v infinite and k,Ī»<v is resolvable and, in fact, has Ī± orthogonal resolutions for each Ī±<v. We also show that, while a t-(v,k,Ī) design with t and Ī» finite, v infinite and k=v may or may not have a resolution, any resolution of such a design must have v parallel classes containing v blocks and at most Ī»ā1 parallel classes containing fewer than v blocks. Further, a resolution into parallel classes of any specified sizes obeying these conditions is realisable in some design. When k<v and Ī»=v and when k=v and Ī» is infinite, we give various examples of resolvable and non-resolvable t-(v,k,Ī) designs
Properties of Steiner triple systems of order 21
Properties of the 62,336,617 Steiner triple systems of order 21 with a
non-trivial automorphism group are examined. In particular, there are 28 which
have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20
that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and
two that avoid the prism. All systems contain the grid. None have a block
intersection graph that is 3-existentially closed.Comment: 12 page
Generalized packing designs
Generalized -designs, which form a common generalization of objects such
as -designs, resolvable designs and orthogonal arrays, were defined by
Cameron [P.J. Cameron, A generalisation of -designs, \emph{Discrete Math.}\
{\bf 309} (2009), 4835--4842]. In this paper, we define a related class of
combinatorial designs which simultaneously generalize packing designs and
packing arrays. We describe the sometimes surprising connections which these
generalized designs have with various known classes of combinatorial designs,
including Howell designs, partial Latin squares and several classes of triple
systems, and also concepts such as resolvability and block colouring of
ordinary designs and packings, and orthogonal resolutions and colourings.
Moreover, we derive bounds on the size of a generalized packing design and
construct optimal generalized packings in certain cases. In particular, we
provide methods for constructing maximum generalized packings with and
block size or 4.Comment: 38 pages, 2 figures, 5 tables, 2 appendices. Presented at 23rd
British Combinatorial Conference, July 201
- ā¦