  The boxicity (respectively cubicity) of a graph $G$ is the minimum
non-negative integer $k$, such that $G$ can be represented as an intersection
graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit
cubes) and is denoted by $box(G)$ (respectively $cub(G)$). It was shown by
Adiga and Chandran (Journal of Graph Theory, 65(4), 2010) that for any graph
$G$, $cub(G) \le$ box$(G) \left \lceil \log_2 \alpha \right \rceil$, where
$\alpha = \alpha(G)$ is the cardinality of the maximum independent set in $G$.
In this note we show that $cub(G) \le 2 \left \lceil \log_2 \chi(G) \right
\rceil box(G) + \chi(G) \left \lceil \log_2 \alpha(G) \right \rceil $. In
general, this result can provide a much better upper bound than that of Adiga
and Chandran for graph classes with bounded chromatic number. For example, for
bipartite graphs we get, $cub(G) \le 2 (box(G) + \left \lceil \log_2 \alpha(G)
\right \rceil )$.
  Moreover we show that for every positive integer $k$, there exist graphs with
chromatic number $k$, such that for every $\epsilon > 0$, the value given by
our upper bound is at most $(1+\epsilon)$ times their cubicity. Thus, our upper
bound is almost tight.
