63,246 research outputs found
More nonexistence results for symmetric pair coverings
A -covering is a pair , where is a
-set of points and is a collection of -subsets of
(called blocks), such that every unordered pair of points in is contained
in at least blocks in . The excess of such a covering is
the multigraph on vertex set in which the edge between vertices and
has multiplicity , where is the number of blocks which
contain the pair . A covering is symmetric if it has the same number
of blocks as points. Bryant et al.(2011) adapted the determinant related
arguments used in the proof of the Bruck-Ryser-Chowla theorem to establish the
nonexistence of certain symmetric coverings with -regular excesses. Here, we
adapt the arguments related to rational congruence of matrices and show that
they imply the nonexistence of some cyclic symmetric coverings and of various
symmetric coverings with specified excesses.Comment: Submitted on May 22, 2015 to the Journal of Linear Algebra and its
Application
On the job rotation problem
The job rotation problem (JRP) is the following: Given an matrix over \Re \cup \{\ -\infty\ \}\ and , find a principal submatrix of whose optimal assignment problem value is maximum. No polynomial algorithm is known for solving this problem if is an input variable. We analyse JRP and present polynomial solution methods for a number of special cases
Parity of the spin structure defined by a quadratic differential
According to the work of Kontsevich-Zorich, the invariant that classifies
non-hyperelliptic connected components of the moduli spaces of Abelian
differentials with prescribed singularities,is the parity of the spin
structure.
We show that for the moduli space of quadratic differentials, the spin
structure is constant on every stratum where it is defined. In particular this
disproves the conjecture that it classifies the non-hyperelliptic connected
components of the strata of quadratic differentials with prescribed
singularities. An explicit formula for the parity of the spin structure is
given.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper12.abs.htm
Counting dimer coverings on self-similar Schreier graphs
We study partition functions for the dimer model on families of finite graphs
converging to infinite self-similar graphs and forming approximation sequences
to certain well-known fractals. The graphs that we consider are provided by
actions of finitely generated groups by automorphisms on rooted trees, and thus
their edges are naturally labeled by the generators of the group. It is thus
natural to consider weight functions on these graphs taking different values
according to the labeling. We study in detail the well-known example of the
Hanoi Towers group , closely related to the Sierpi\'nski gasket.Comment: 29 pages. Final version, to appear in European Journal of
Combinatoric
A Note on the Sparing Number of Graphs
An integer additive set-indexer is defined as an injective function
such that the induced function defined by is also
injective. An IASI is said to be a weak IASI if
for all . A graph which admits a
weak IASI may be called a weak IASI graph. The set-indexing number of an
element of a graph , a vertex or an edge, is the cardinality of its
set-labels. The sparing number of a graph is the minimum number of edges
with singleton set-labels, required for a graph to admit a weak IASI. In
this paper, we study the sparing number of certain graphs and the relation of
sparing number with some other parameters like matching number, chromatic
number, covering number, independence number etc.Comment: 10 pages, 10 figures, submitte
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