Packing coloring of graphs with long paths

TL;DR

This paper extends the understanding of packing coloring in graphs with long paths by introducing path-aligned graph products, analyzing their packing chromatic numbers, and characterizing caterpillars with low packing chromatic numbers.

Contribution

It introduces path-aligned graph products, proves bounded packing chromatic numbers for these families, and characterizes caterpillars with packing chromatic number at most 3.

Findings

Packing chromatic number is bounded by a constant for certain path-aligned graph families.

Complete structural characterization of caterpillars with packing chromatic number ≤ 3.

Extended results on packing chromatic numbers beyond paths and cycles.

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.