  The b-chromatic number $b(G)$ of a graph $G$ is the maximum $k$ for which $G$
has a proper vertex coloring using $k$ colors such that each color class
contains at least one vertex adjacent to a vertex of every other color class.
In this paper, we mainly investigate on one of the open problems given in [P.
Francis, S. Francis Raj, On b-coloring of powers of hypercubes, Discrete Appl.
Math. 225 (2017) 74-86.]. As a consequence, we have obtained an upper bound for
the b-chromatic number of some powers of hypercubes. This turns out to be an
improvement of the already existing bound in [P. Francis, S. Francis Raj, On
b-coloring of powers of hypercubes, Discrete Appl. Math. 225 (2017) 74-86.].
Further, we have determined a lower bound for the b-chromatic number of some
powers of the Hamming graph, a generalization of the hypercube.
