7 research outputs found
Rainbow Thresholds
We extend a recent breakthrough result relating expectation thresholds and
actual thresholds to include rainbow versions
The -visibility Localization Game
We study a variant of the Localization game in which the cops have limited
visibility, along with the corresponding optimization parameter, the
-visibility localization number , where is a non-negative
integer. We give bounds on -visibility localization numbers related to
domination, maximum degree, and isoperimetric inequalities. For all , we
give a family of trees with unbounded values. Extending results known
for the localization number, we show that for , every tree contains a
subdivision with . For many , we give the exact value of
for the Cartesian grid graphs, with the remaining cases
being one of two values as long as is sufficiently large. These examples
also illustrate that for all distinct choices of and
$j.
Iterative models for complex networks formed by extending cliques
We consider a new model for complex networks whose underlying mechanism is
extending dense subgraphs. In the frustum model, we iteratively extend cliques
over discrete-time steps. For many choices of the underlying parameters, graphs
generated by the model densify over time. In the special case of the cone
model, generated graphs provably satisfy properties observed in real-world
complex networks such as the small world property and bad spectral expansion.
We finish with a set of open problems and next steps for the frustum model
The enumeration of cyclic mutually nearly orthogonal Latin squares
In this paper, we study collections of mutually nearly orthogonal Latin squares (MNOLS), which come from a modification of the orthogonality condition for mutually orthogonal Latin squares. In particular, we find the maximum μ such that there exists a set of μ cyclic MNOLS of order n for n≤18, as well as providing a full enumeration of sets and lists of μ cyclic MNOLS of order n under a variety of equivalences with n≤18. This resolves in the negative a conjecture that proposed that the maximum μ for which a set of μ cyclic MNOLS of order n exists is [n/4]+1
Computing autotopism groups of partial Latin rectangles: A pilot study
Computing the autotopism group of a partial Latin rectangle can be performed in a variety
of ways. This pilot study has two aims: (a) to compare these methods experimentally, and (b) to
identify the design goals one should have in mind for developing practical software. To this end,
we compare six families of algorithms (two backtracking methods and four graph automorphism
methods), with and without the use of entry invariants, on two test suites. We consider two
entry invariants: one determined by the frequencies of row, column, and symbol representatives,
and one determined by 2 2 submatrices. We nd: (a) with very few entries, many symmetries
often exist, and these should be identi ed mathematically rather than computationally, (b) with
an intermediate number of entries, a quick-to-compute entry invariant was e ective at reducing
the need for computation, (c) with an almost-full partial Latin rectangle, more sophisticated
entry invariants are needed, and (d) the performance for (full) Latin squares is signi cantly
poorer than other partial Latin rectangles of comparable size, obstructed by the existence of
Latin squares with large (possibly transitive) autotopism groups.Junta de AndalucÃa FQM-01