Packing coloring of graphs with long paths

Abstract: The packing coloring problem has diverse applications, including frequency assignment in wireless networks, resource distribution and facility location in smart cities and post-disaster management, as well as in biological diversity. Formally, the packing coloring of a graph is a vertex coloring in which any two vertices assigned color $i$ are at a distance of at least $i+1$, and the smallest number of colors admitting such a coloring is called the packing chromatic number. Goddard et al.~\cite{goddard2008broadcast} showed that the packing chromatic numbers of paths and cycles are at most 3 and 4, respectively. In this paper, we introduce \emph{path-aligned graph products}, a natural extension of paths with unbounded diameter. We extend the result of~\cite{goddard2008broadcast} by proving that the packing chromatic number remains bounded by a constant for several families of path-aligned cycle and path-aligned complete products. We then investigate the packing chromatic number of caterpillars, another class of graphs characterized by long induced paths. Sloper~\cite{sloper} proved that the packing chromatic number of caterpillars is at most 7; here, we provide a complete structural characterization of caterpillars with packing chromatic number at most 3. Finally, several open research questions are posed.