1,470 research outputs found
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
Recommended from our members
On the number of additive permutations and Skolem-type sequences
Cavenagh and Wanless recently proved that, for sufficiently large odd n, the number of transversals in the Latin square formed from the addition table for integers modulo n is greater than (3.246)n. We adapt their proof to show that for sufficiently large t the number of additive permutations on [-t,t] is greater than (3.246)2t+1 and we go on to derive some much improved lower bounds on the numbers of Skolem-type sequences. For example, it is shown that for sufficiently large t ≡ 0$ or 3 (mod 4), the number of split Skolem sequences of order n=7t+3 is greater than (3.246)6t+3. This compares with the previous best bound of 2⌊n/3⌋
On the number of transversals in latin squares
The logarithm of the maximum number of transversals over all latin squares of
order is greater than
Hypergraphs and hypermatrices with symmetric spectrum
It is well known that a graph is bipartite if and only if the spectrum of its
adjacency matrix is symmetric. In the present paper, this assertion is
dissected into three separate matrix results of wider scope, which are extended
also to hypermatrices. To this end the concept of bipartiteness is generalized
by a new monotone property of cubical hypermatrices, called odd-colorable
matrices. It is shown that a nonnegative symmetric -matrix has a
symmetric spectrum if and only if is even and is odd-colorable. This
result also solves a problem of Pearson and Zhang about hypergraphs with
symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu.
Separately, similar results are obtained for the -spectram of
hypermatrices.Comment: 17 pages. Corrected proof on p. 1
Chromatic Numbers of Simplicial Manifolds
Higher chromatic numbers of simplicial complexes naturally
generalize the chromatic number of a graph. In any fixed dimension
, the -chromatic number of -complexes can become arbitrarily
large for [6,18]. In contrast, , and only
little is known on for .
A particular class of -complexes are triangulations of -manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high were known.
We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere with face vector
and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
On some points-and-lines problems and configurations
We apply an old method for constructing points-and-lines configurations in
the plane to study some recent questions in incidence geometry.Comment: 14 pages, numerous figures of point-and-line configurations; to
appear in the Bezdek-50 special issue of Periodica Mathematica Hungaric
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