The packing chromatic number $\chi_{\pi}(G)$ of a graph $G$ is the smallest integer $k$ for which a packing k-coloring of graph $G$ can be found, which is the smallest $k$ for which such a function $\pi: (G) \to [k]$ exists, that from $\pi(u) = \pi(v)$ follows that the distance between u and v is greater than $\pi(u)$. For a graph $G$ and $p \ge 1$, a $p$-coronae of the graph $G$ is defined as the graph we obtain graph $G$ by adding p additional leaves (vertices of degree 1) to each vertex on the graph.
Determining the packing chromatic number of a graph is a complex problem. In this paper we show this by presenting a proof that 4-packing coloring is an NP-complete problem. Then we prove a theorem on the packing chromatic number of paths and cycles, and afterwards focus on the packing chromatic number of $p$-coronae of paths and cycles.
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