3,917 research outputs found
A Note On Transversals
Let be a finite group and a core-free subgroup of . We will show
that if there exists a solvable, generating transversal of in , then
is a solvable group. Further, if is a generating transversal of in
and has order 2 invariant sub right loop such that the quotient
is a group. Then is an elementary abelian 2-group.Comment: 7 page
A note on full transversals and mixed orthogonal arrays
We investigate a packing problem in M-dimensional grids, where bounds are given for the number of allowed entries in different axis-parallel directions. The concept is motivated from error correcting codes and from more-part Sperner theory. It is also closely related to orthogonal arrays. We prove that some packing always reaches the natural upper bound for its size, and even more, one can partition the grid into such packings, if a necessary divisibility condition holds. We pose some extremal problems on maximum size of packings, such that packings of that size always can be extended to meet the natural upper bound. 1 The concept of full transversals Let us be given positive integers n1,n2,...,nM and L1,L2,...,LM, such tha
On the number of -transversals in hypergraphs
A set of vertices in a hypergraph is \textit{strongly independent} if
every hyperedge shares at most one vertex with . We prove a sharp result for
the number of maximal strongly independent sets in a -uniform hypergraph
analogous to the Moon-Moser theorem.
Given an -uniform hypergraph and a non-empty set of
non-negative integers, we say that a set is an \textit{-transversal} of
if for any hyperedge of , we have
\mbox{}. Independent sets are
-transversals, while strongly independent sets are
-transversals. Note that for some sets , there may exist
hypergraphs without any -transversals. We study the maximum number of
-transversals for every , but we focus on the more natural sets, e.g.,
, or being the set of odd or the set of even
numbers.Comment: 10 page
Note on a theorem of J. Folkman on transversals of infinite families with finitely many infinite members
In this note we show by a simple direct proof that Folkman's necessary and sufficient condition for an infinite family of sets with finitely many infinite members to have a transversal implies Woodall's condition. A short proof of Folkman's theorem results by combining with Woodall's proof
Parity of transversals of Latin squares
We introduce a notion of parity for transversals, and use it to show that in
Latin squares of order , the number of transversals is a multiple of
4. We also demonstrate a number of relationships (mostly congruences modulo 4)
involving , where is the number of diagonals of a given
Latin square that contain exactly different symbols.
Let denote the matrix obtained by deleting row and column
from a parent matrix . Define to be the number of transversals
in , for some fixed Latin square . We show that for all and . Also, if has odd order then the
number of transversals of equals mod 2. We conjecture that for all .
In the course of our investigations we prove several results that could be of
interest in other contexts. For example, we show that the number of perfect
matchings in a -regular bipartite graph on vertices is divisible by
when is odd and . We also show that for all , when is an integer matrix of odd
order with all row and columns sums equal to
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