  A b-coloring of a graph $G$ is a coloring of its vertices such that every
color class contains a vertex that has neighbors in all other classes. The
b-chromatic number of $G$ is the largest integer $k$ such that $G$ has a
b-coloring with $k$ colors. Javadi and Omoomi ("On b-coloring of cartesian
product of graphs", Ars Combinatoria 107 (2012) 521-536) proved that the
b-chromatic number of $K_n \times K_n$ (the Cartesian product of two complete
graphs on $n$ vertices) is in the set $\{2n-3, 2n-2\}$ and conjectured that the
exact value is $2n-3$ for all $n \ge 5$. We give counterexamples to this
conjecture for $n=5$, $n=6$ and $n=7$.
