  A vertex $v$ of a given graph $G$ is said to be in a rainbow neighbourhood of
$G$, with respect to a proper coloring $C$ of $G$, if the closed neighbourhood
$N[v]$ of the vertex $v$ consists of at least one vertex from every colour
class of $G$ with respect to $C$. A maximal proper colouring of a graph $G$ is
a $J$-colouring of $G$ if and only if every vertex of G belongs to a rainbow
neighbourhood of $G$. In this paper, we study certain parameters related to
$J$-colouring of certain Mycielski type graphs.
