191 research outputs found
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
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
Block-avoiding point sequencings
Let and be positive integers. Recent papers by Kreher, Stinson and
Veitch have explored variants of the problem of ordering the points in a triple
system (such as a Steiner triple system, directed triple system or Mendelsohn
triple system) on points so that no block occurs in a segment of
consecutive entries (thus the ordering is locally block-avoiding). We describe
a greedy algorithm which shows that such an ordering exists, provided that
is sufficiently large when compared to . This algorithm leads to improved
bounds on the number of points in cases where this was known, but also extends
the results to a significantly more general setting (which includes, for
example, orderings that avoid the blocks of a design). Similar results for a
cyclic variant of this situation are also established.
We construct Steiner triple systems and quadruple systems where can be
large, showing that a bound of Stinson and Veitch is reasonable. Moreover, we
generalise the Stinson--Veitch bound to a wider class of block designs and to
the cyclic case.
The results of Kreher, Stinson and Veitch were originally inspired by results
of Alspach, Kreher and Pastine, who (motivated by zero-sum avoiding sequences
in abelian groups) were interested in orderings of points in a partial Steiner
triple system where no segment is a union of disjoint blocks. Alspach~\emph{et
al.}\ show that, when the system contains at most pairwise disjoint blocks,
an ordering exists when the number of points is more than . By making
use of a greedy approach, the paper improves this bound to .Comment: 38 pages. Typo in the statement of Theorem 17 corrected, and other
minor changes mad
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Combinatorial Embeddings and Representations
Topological embeddings of complete graphs and complete multipartite graphs give rise to combinatorial designs when the faces of the embeddings are triangles. In this case, the blocks of the design correspond to the triangular faces of the embedding. These designs include Steiner, twofold and Mendelsohn triple systems, as well as Latin squares. We look at construction methods, structural properties and other problems concerning these cases.
In addition, we look at graph representations by Steiner triple systems and by combinatorial embeddings. This is closely related to finding independent sets in triple systems. We examine which graphs can be represented in Steiner triple systems and combinatorial embeddings of small orders and give several bounds including a bound on the order of Steiner triple systems that are guaranteed to represent all graphs of a given maximum degree. Finally, we provide an enumeration of graphs of up to six edges representable by Steiner triple systems
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given
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