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 vertices in $V_i$ are pairwise at distance at least $i+1$. Packing chromatic vertex-critical graphs, $\chi_{ \rho}$-critical for short, are introduced as the graphs $G$ for which $\chi_{ \rho}(G-x) < \chi_{ \rho}(G)$ holds for every vertex $x$ of $G$. If $\chi_{ \rho}(G) = k$, then $G$ is $k$-$\chi_{ \rho}$-critical. It is shown that if $G$ is $\chi_{ \rho}$-critical, then the set $\{ \chi_{ \rho}(G) - \chi_{ \rho}(G-x):\ x\in V(G)\}$ can be almost arbitrary. The $3$-$\chi_{ \rho}$-critical graphs are characterized, and $4$-$\chi_{ \rho}$-critical graphs are characterized in the case when they contain a cycle of length at least $5$ which is not congruent to $0$ modulo $4$. It is shown that for every integer $k\ge 2$ there exists a $k$-$\chi_{ \rho}$-critical tree and that a $k$-$\chi_{ \rho}$-critical caterpillar exists if and only if $k\le 7$. Cartesian products are also considered and in particular it is proved that if $G$ and $H$ are vertex-transitive graphs and ${ \rm diam(G)} + { \rm diam}(H) \le \chi_{ \rho}(G)$, then $G\,\square\, H$ is $\chi_{ \rho}$-critical.
Categories: math.CO