[Paper Review] On the Packing Chromatic Number on Hamming Graphs and General Graphs

Graciela Nasini, Daniel Severín+1 more|arXiv (Cornell University)|Oct 19, 2015

Graph Labeling and Dimension Problems6 references3 citations

TL;DR

This paper establishes a direct formula for the packing chromatic number (PCN) of Hamming graphs $H_{q,m}$ when $m=2,3$, and provides improved upper bounds for $m \geq 4$ using integer linear programming. It also introduces a polynomial-time reduction from the PCN problem to the Maximum Stable Set (MSS) problem via a constructed graph $G_*^{[d-1]}$, enabling efficient computation of PCN through MSS solvers, with significant improvements for hypercubes up to dimension 13.

ABSTRACT

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ needed to proper color the vertices of $G$ in such a way the distance between any two vertices having color $i$ be at least $i+1$. We obtain $χ_ρ(H_{q,m})$ for $m=3$, where $H_{q,m}$ is the Hamming graph of words of length $m$ and alphabet with $q$ symbols, and tabulate bounds of them for $m \geq 4$ up to 10000 vertices. We also give a polynomial reduction from the problem of finding $χ_ρ(G)$ to the Maximum Stable Set problem.

Motivation & Objective

• To determine the exact packing chromatic number $\chi_{\rho}(H_{q,m})$ for Hamming graphs with $m=2,3$ and arbitrary $q$.
• To compute tight upper bounds for $\chi_{\rho}(H_{q,m})$ for $m \geq 4$ up to 10,000 vertices using integer linear programming.
• To establish a polynomial-time reduction from the packing chromatic number problem to the Maximum Stable Set (MSS) problem.
• To enable efficient computation of $\chi_{\rho}(G)$ for graphs where previous methods were inapplicable, especially for hypercubes and Hamming graphs with $q \geq 3$.

Proposed method

• Derive a closed-form expression for $\chi_{\rho}(H_{q,m})$ using induction and the known lower bound from Brešar et al. (2007), proving equality for $m=2,3$.
• Apply Lemma 1.1 to express $\chi_{\rho}(H_{q,m}) = m-1 + q^m - \alpha(H_{q,m}^{[m-1]})$, reducing the problem to computing the stability number of $H_{q,m}^{[m-1]}$.
• Use integer linear programming to compute upper bounds on $\chi_{\rho}(H_{q,m})$ for $m \geq 4$, leveraging the relationship between PCN and stability numbers.
• Construct a new graph $G_*^{[d-1]}$ from $G^{[d-1]}$ by adding a universal vertex $*$ connected to all $(v,k)$ vertices, enabling a polynomial reduction to MSS.
• Prove that $\chi_{\rho}(G) = (d-1) + n - \alpha(G_*^{[d-1]})$, allowing exact computation of PCN via MSS solvers.
• Utilize the reduction to improve upper bounds for hypercubes: $\chi_{\rho}(H_{2,11}) \leq 857$ (down from 881), and compute $\chi_{\rho}(H_{2,12}) \leq 1707$, $\chi_{\rho}(H_{2,13}) \leq 3641$.

Experimental results

Research questions

• RQ1 What is the exact value of the packing chromatic number $\chi_{\rho}(H_{q,m})$ for $m=2$ and $m=3$ for any $q \geq 2$?
• RQ2 Can tight upper bounds be computed for $\chi_{\rho}(H_{q,m})$ when $m \geq 4$ and the number of vertices is up to 10,000?
• RQ3 Is there a polynomial-time reduction from the packing chromatic number problem to the Maximum Stable Set problem?
• RQ4 Can the proposed reduction be used to compute or improve bounds on $\chi_{\rho}(G)$ for graphs like hypercubes and Hamming graphs with $q \geq 3$ where prior results are limited?

Key findings

• For $m=2$ and $m=3$, the paper establishes that $\chi_{\rho}(H_{q,m}) = m-1 + q^m - \alpha(H_{q,m}^{[m-1]})$, and this formula holds as equality.
• The paper improves the upper bound for the packing chromatic number of the 11-dimensional hypercube from 881 to 857 using integer linear programming.
• For the 12-dimensional hypercube, the upper bound is computed as $\chi_{\rho}(H_{2,12}) \leq 1707$, and for the 13-dimensional hypercube, $\chi_{\rho}(H_{2,13}) \leq 3641$.
• A polynomial-time reduction is established: $\chi_{\rho}(G) = (d-1) + n - \alpha(G_*^{[d-1]})$, where $G_*^{[d-1]}$ is a graph constructed by adding a universal vertex to $G^{[d-1]}$.
• The reduction allows the use of MSS solvers to compute or bound $\chi_{\rho}(G)$, especially useful for graphs where $\alpha(G^{[k]})$ is hard to compute directly.
• The method enables the computation of $\chi_{\rho}(G)$ for graphs with diameter $d \geq 2$ by reducing the problem to computing the stability number of a graph of size $(n+1)(d-1)$.