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Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

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$(G) \left \lceil \log_2 α\right \rceil$, where $α= α(G)$ is the cardinality of the maximum independent set in $G$. In this note we show that $cub(G) \le 2 \left \lceil \log_2 χ(G) \right \rceil box(G) + χ(G) \left \lceil \log_2 α(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, $cub(G) \le 2 (box(G) + \left \lceil \log_2 α(G) \right \rceil )$. Moreover we show that for every positive integer $k$, there exist graphs with chromatic number $k$, such that for every $ε> 0$, the value given by our upper bound is at most $(1+ε)$ times their cubicity. Thus, our upper bound is almost tight.