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Packing coloring of graphs with long paths

Hanna Furmańczyk, Didem Gözüpek, Sibel Özkan

TL;DR

This work introduces path-aligned graph products $P_n \Diamond_{\ell} G$, extending paths by attaching copies of a vertex-transitive graph along segments and investigates their packing chromatic numbers $\chi_p$. It establishes that $\chi_p$ remains bounded by a constant for several path-aligned cycle and complete products, highlighting cases where these otherwise hard problems become tractable within this graph family. The paper also advances the theory for caterpillars by providing a tight structural characterization of those with $\chi_p(CT_l)=3$ and improving general upper bounds on $\chi_p$ for caterpillars, including a complete classification via seven explicit families. Together, these results yield both new polynomial-time recognition insights for specific graph classes and concrete coloring constructions (often via explicit patterns) for a broad spectrum of path-aligned structures, with open questions guiding future work on larger complete graphs and broader cycle families.

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.

Packing coloring of graphs with long paths

TL;DR

This work introduces path-aligned graph products , extending paths by attaching copies of a vertex-transitive graph along segments and investigates their packing chromatic numbers . It establishes that remains bounded by a constant for several path-aligned cycle and complete products, highlighting cases where these otherwise hard problems become tractable within this graph family. The paper also advances the theory for caterpillars by providing a tight structural characterization of those with and improving general upper bounds on for caterpillars, including a complete classification via seven explicit families. Together, these results yield both new polynomial-time recognition insights for specific graph classes and concrete coloring constructions (often via explicit patterns) for a broad spectrum of path-aligned structures, with open questions guiding future work on larger complete graphs and broader cycle families.

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 are at a distance of at least , 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.

Paper Structure

This paper contains 13 sections, 31 theorems, 36 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Path-aligned graph products
  4. Path-aligned cycle products
  5. Cycles $C_{4s}$ for $s \geq 1$
  6. Cycles $C_{4s+1}$ for $s\geq 1$
  7. Cycles $C_{4s+2}$ for $s\geq 1$
  8. Cycles $C_{4s+3}$ for $s\geq 0$
  9. Summarizing remarks
  10. Path-aligned complete products
  11. Summarizing remarks
  12. Packing Coloring of Caterpillars
  13. Characterization of caterpillars with $\chi_p(CT_l)=3$

Key Result

Proposition 1

$\chi_{\rho}(P_n) = $

Figures (5)

  • Figure 1: 5-packing coloring of $P_{10} \mathbin{\Diamond_2} C_4$.
  • Figure 2: Packing 5-coloring of $P_{10} \mathbin{\Diamond_2} C_8$.
  • Figure 3: Packing 4-coloring of $P_{15} \mathbin{\Diamond_3} C_8$.
  • Figure 4: Packing 6-coloring of $P_{6} \mathbin{\Diamond_2} K_4$.
  • Figure 5: The caterpillar $C(4;4,1,0,1)$ with the backbone $v_1v_2v_3v_4$ of length 4.

Theorems & Definitions (58)

  • Proposition 1: goddard2008broadcast
  • Proposition 2: goddard2008broadcast
  • Proposition 3: goddard2008broadcast
  • Proposition 4
  • proof
  • Definition 1: GoRo13
  • Definition 2
  • Proposition 5
  • proof
  • Theorem 6
  • ...and 48 more