  A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of
non-negative integers (with minimum assignment $0$) to its vertices such that
the adjacent vertices must receive integers at least two apart and vertices at
distance two must receive distinct integers. The lambda chromatic number (or
the $\lambda$ number) of a graph $G$ is the least positive integer among all
the maximum assigned positive integer over all possible lambda colouring of the
graph $G$. Here we have primarily shown that every graph with lambda chromatic
number $t$ can be embedded in a graph, with lambda chromatic number $t$, which
admits a partition of the vertex set into colour classes of equal size. It is
further proved that if an $n-$vertex graph with lambda chromatic number
$t\geq5$, where $n\geq t+1$, contains maximum number of edges, then the vertex
set of such graph admits an equitable partition. For such an admitted equitable
partition there are either $0$ or $\min\{|A|,|B|\}$ number of edges between
each pair $(A,B)$ of subsets (i.e. roughly, such partition is a "sparse like"
equitable partition). Here we establish a classification result, identifying
all possible $n-$vertex graphs with lambda chromatic number $t\geq3$, where
$n\geq t+1$, which contain maximum number of edges. Such classification
provides a solution of a problem posed more than two decades ago by John P.
Georges and David W. Mauro.
