  The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest
integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$,
$i\in [k]$, where each $V_i$ is an $i$-packing. In this paper, we investigate
for a given triple $(a,b,c)$ of positive integers whether there exists a graph
$G$ such that $\omega(G) = a$, $\chi(G) = b$, and $\chi_{\rho}(G) = c$. If so,
we say that $(a, b, c)$ is realizable. It is proved that $b=c\ge 3$ implies
$a=b$, and that triples $(2,k,k+1)$ and $(2,k,k+2)$ are not realizable as soon
as $k\ge 4$. Some of the obtained results are deduced from the bounds proved on
the packing chromatic number of the Mycielskian. Moreover, a formula for the
independence number of the Mycielskian is given. A lower bound on
$\chi_{\rho}(G)$ in terms of $\Delta(G)$ and $\alpha(G)$ is also proved.
