171 research outputs found
On Rainbow Connection Number and Connectivity
Rainbow connection number, , of a connected graph is the minimum
number of colours needed to colour its edges, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we investigate the relationship of rainbow connection number with
vertex and edge connectivity. It is already known that for a connected graph
with minimum degree , the rainbow connection number is upper bounded by
[Chandran et al., 2010]. This directly gives an upper
bound of and for rainbow
connection number where and , respectively, denote the edge
and vertex connectivity of the graph. We show that the above bound in terms of
edge connectivity is tight up-to additive constants and show that the bound in
terms of vertex connectivity can be improved to , for any . We conjecture that rainbow connection
number is upper bounded by and show that it is true for
. We also show that the conjecture is true for chordal graphs and
graphs of girth at least 7.Comment: 10 page
TinkerCell: Modular CAD Tool for Synthetic Biology
Synthetic biology brings together concepts and techniques from engineering
and biology. In this field, computer-aided design (CAD) is necessary in order
to bridge the gap between computational modeling and biological data. An
application named TinkerCell has been created in order to serve as a CAD tool
for synthetic biology. TinkerCell is a visual modeling tool that supports a
hierarchy of biological parts. Each part in this hierarchy consists of a set of
attributes that define the part, such as sequence or rate constants. Models
that are constructed using these parts can be analyzed using various C and
Python programs that are hosted by TinkerCell via an extensive C and Python
API. TinkerCell supports the notion of a module, which are networks with
interfaces. Such modules can be connected to each other, forming larger modular
networks. Because TinkerCell associates parameters and equations in a model
with their respective part, parts can be loaded from databases along with their
parameters and rate equations. The modular network design can be used to
exchange modules as well as test the concept of modularity in biological
systems. The flexible modeling framework along with the C and Python API allows
TinkerCell to serve as a host to numerous third-party algorithms. TinkerCell is
a free and open-source project under the Berkeley Software Distribution
license. Downloads, documentation, and tutorials are available at
www.tinkercell.com.Comment: 23 pages, 20 figure
Boxicity and Cubicity of Product Graphs
The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such
that G can be represented as an intersection graph of axis-parallel rectangular
boxes (axis-parallel unit cubes) in . In this article, we give estimates
on the boxicity and the cubicity of Cartesian, strong and direct products of
graphs in terms of invariants of the component graphs. In particular, we study
the growth, as a function of , of the boxicity and the cubicity of the
-th power of a graph with respect to the three products. Among others, we
show a surprising result that the boxicity and the cubicity of the -th
Cartesian power of any given finite graph is in and
, respectively. On the other hand, we show that there
cannot exist any sublinear bound on the growth of the boxicity of powers of a
general graph with respect to strong and direct products.Comment: 14 page
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