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

Boxicity and Cubicity of Asteroidal Triple free graphs

An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as \boxi(G), is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denoted as \cub(G), is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-cubes. Let $S(m)$ denote a star graph on $m+1$ nodes. We define {\it claw number} of a graph $G$ as the largest positive integer $k$ such that $S(k)$ is an induced subgraph of $G$ and denote it as \claw. Let $G$ be an AT-free graph with chromatic number $\chi(G)$ and claw number \claw. In this paper we will show that \boxi(G) \leq \chi(G) and this bound is tight. We also show that \cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2) $\leq$ \chi(G)(\ceil{\log_2 \claw} +2). If $G$ is an AT-free graph having girth at least 5 then \boxi(G) \leq 2 and therefore \cub(G) \leq 2\ceil{\log_2 \claw} +4.Comment: 15 pages: We are replacing our earlier paper regarding boxicity of permutation graphs with a superior result. Here we consider the boxicity of AT-free graphs, which is a super class of permutation graph

Representing a cubic graph as the intersection graph of axis-parallel boxes in three dimensions

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes just touch at their boundaries. Further, this construction can be realized in linear time

Cubicity of interval graphs and the claw number

Let $G(V,E)$ be a simple, undirected graph where $V$ is the set of vertices and $E$ is the set of edges. A $b$-dimensional cube is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval of unit length on the real line. The \emph{cubicity} of $G$, denoted by \cub(G) is the minimum positive integer $b$ such that the vertices in $G$ can be mapped to axis parallel $b$-dimensional cubes in such a way that two vertices are adjacent in $G$ if and only if their assigned cubes intersect. Suppose $S(m)$ denotes a star graph on $m+1$ nodes. We define \emph{claw number} $\psi(G)$ of the graph to be the largest positive integer $m$ such that $S(m)$ is an induced subgraph of $G$. It can be easily shown that the cubicity of any graph is at least \ceil{\log_2\psi(G)}. In this paper, we show that, for an interval graph $G$ \ceil{\log_2\psi(G)}\le\cub(G)\le\ceil{\log_2\psi(G)}+2. Till now we are unable to find any interval graph with \cub(G)>\ceil{\log_2\psi(G)}. We also show that, for an interval graph $G$, \cub(G)\le\ceil{\log_2\alpha}, where $\alpha$ is the independence number of $G$. Therefore, in the special case of $\psi(G)=\alpha$, \cub(G) is exactly \ceil{\log_2\alpha}. The concept of cubicity can be generalized by considering boxes instead of cubes. A $b$-dimensional box is a Cartesian product $I_1\times I_2\times...\times I_b$, where each $I_i$ is a closed interval on the real line. The \emph{boxicity} of a graph, denoted $box(G)$, is the minimum $k$ such that $G$ is the intersection graph of $k$-dimensional boxes. It is clear that box(G)\le\cub(G). From the above result, it follows that for any graph $G$, \cub(G)\le box(G)\ceil{\log_2\alpha}

Bipartite powers of k-chordal graphs

Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G^m is chordal then so is G^{m+2}. Brandst\"adt et al. in [Andreas Brandst\"adt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics, 177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}. Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. Given a bipartite graph G and an odd positive integer m, we define the graph G^{[m]} to be a bipartite graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power of G. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is odd.Comment: 10 page