13,550 research outputs found
List Distinguishing Parameters of Trees
A coloring of the vertices of a graph G is said to be distinguishing}
provided no nontrivial automorphism of G preserves all of the vertex colors.
The distinguishing number of G, D(G), is the minimum number of colors in a
distinguishing coloring of G. The distinguishing chromatic number of G,
chi_D(G), is the minimum number of colors in a distinguishing coloring of G
that is also a proper coloring.
Recently the notion of a distinguishing coloring was extended to that of a
list distinguishing coloring. Given an assignment L= {L(v) : v in V(G)} of
lists of available colors to the vertices of G, we say that G is (properly)
L-distinguishable if there is a (proper) distinguishing coloring f of G such
that f(v) is in L(v) for all v. The list distinguishing number of G, D_l(G), is
the minimum integer k such that G is L-distinguishable for any list assignment
L with |L(v)| = k for all v. Similarly, the list distinguishing chromatic
number of G, denoted chi_{D_l}(G) is the minimum integer k such that G is
properly L-distinguishable for any list assignment L with |L(v)| = k for all v.
In this paper, we study these distinguishing parameters for trees, and in
particular extend an enumerative technique of Cheng to show that for any tree
T, D_l(T) = D(T), chi_D(T)=chi_{D_l}(T), and chi_D(T) <= D(T) + 1.Comment: 10 page
Elliptic rook and file numbers
Utilizing elliptic weights, we construct an elliptic analogue of rook numbers
for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's
q-rook numbers by two additional independent parameters a and b, and a nome p.
These are shown to satisfy an elliptic extension of a factorization theorem
which in the classical case was established by Goldman, Joichi and White and
later was extended to the q-case by Garsia and Remmel. We obtain similar
results for our elliptic analogues of Garsia and Remmel's q-file numbers for
skyline boards. We also provide an elliptic extension of the j-attacking model
introduced by Remmel and Wachs. Various applications of our results include
elliptic analogues of (generalized) Stirling numbers of the first and second
kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings
has been taken out to keep the length of this paper reasonable
Predictive Capacity of Meteorological Data - Will it rain tomorrow
With the availability of high precision digital sensors and cheap storage
medium, it is not uncommon to find large amounts of data collected on almost
all measurable attributes, both in nature and man-made habitats. Weather in
particular has been an area of keen interest for researchers to develop more
accurate and reliable prediction models. This paper presents a set of
experiments which involve the use of prevalent machine learning techniques to
build models to predict the day of the week given the weather data for that
particular day i.e. temperature, wind, rain etc., and test their reliability
across four cities in Australia {Brisbane, Adelaide, Perth, Hobart}. The
results provide a comparison of accuracy of these machine learning techniques
and their reliability to predict the day of the week by analysing the weather
data. We then apply the models to predict weather conditions based on the
available data.Comment: 7 pages, 2 Result Set
Distinguishing Number for some Circulant Graphs
Introduced by Albertson et al. \cite{albertson}, the distinguishing number
of a graph is the least integer such that there is a
-labeling of the vertices of that is not preserved by any nontrivial
automorphism of . Most of graphs studied in literature have 2 as a
distinguishing number value except complete, multipartite graphs or cartesian
product of complete graphs depending on . In this paper, we study circulant
graphs of order where the adjacency is defined using a symmetric subset
of , called generator. We give a construction of a family of
circulant graphs of order and we show that this class has distinct
distinguishing numbers and these lasters are not depending on
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