On Rainbow Connection Number and Connectivity

Rainbow connection number, $rc(G)$, of a connected graph $G$ 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 $\delta$, the rainbow connection number is upper bounded by $3n/(\delta + 1) + 3$ [Chandran et al., 2010]. This directly gives an upper bound of $3n/(\lambda + 1) + 3$ and $3n/(\kappa + 1) + 3$ for rainbow connection number where $\lambda$ and $\kappa$, 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 $(2 + \epsilon)n/\kappa + 23/ \epsilon^2$, for any $\epsilon > 0$. We conjecture that rainbow connection number is upper bounded by $n/\kappa + O(1)$ and show that it is true for $\kappa = 2$. 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 $R^k$. 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 $d$, of the boxicity and the cubicity of the $d$-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 $d$-th Cartesian power of any given finite graph is in $O(\log d / \log\log d)$ and $\theta(d / \log d)$, 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