  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
$G$, $\cub(G)\le box(G)\ceil{\log_2\alpha}$.
