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The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallestinteger $k$ such that the vertex set of $G$ can be partitioned into sets$\Pi_1,\ldots,\Pi_k$, where $\Pi_i$, $i\in [k]$, is an $i$-packing. Thefollowing conjecture is posed and studied: if $G$ is a subcubic graph, then$\chi_{\rho}(S(G))\le 5$, where $S(G)$ is the subdivision of $G$. Theconjecture is proved for all generalized prisms of cycles. To get this resultit is proved that if $G$ is a generalized prism of a cycle, then $G$ is$(1,1,2,2)$-colorable if and only if $G$ is not the Petersen graph. Thevalidity of the conjecture is further proved for graphs that can be obtainedfrom generalized prisms in such a way that one of the two $n$-cycles in theedge set of a generalized prism is replaced by a union of cycles among which atmost one is a 5-cycle. The packing chromatic number of graphs obtained bysubdividing each of its edges a fixed number of times is also considered.

Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph