  In this paper, we study the $L(2, 1)$-Labeling of the Mycielski and the
iterated Mycielski of graphs in general. We give sharp bounds for the $L(2,
1)$-number of the $t$th iterated Mycielski of a graph $G$, $\lambda(M^t(G))$
for all $t\geq 1$, in terms of the number of iterations $t$, the order $n$, the
maximum degree, and $\lambda(G)$ the $L(2, 1)$-number of $G$. We present
necessary and sufficient conditions between the $4$-star matching number of the
complement graph and the $L(2, 1)$-number of the Mycielski of a graph
$\lambda(M(G))$. For all $t\geq 2$, we prove that $2^{t-1}(n+2)-2\leq
\lambda(M^t(G))\leq 2^{t}(n+1)-2$ for any graph $G$ of order $n$, then we
completely characterize the graphs achieving the bounds $2^t(n+1)-2$ and
$2^{t-1}(n+2)-2$. We determine the $L(2, 1)$-number for the Mycielski and the
iterated Mycielski of some graph classes.
