Grundy Packing Coloring of Graphs

TL;DR

The paper introduces Grundy packing chromatic number $Γ_{ρ}(G)$ as the maximum output of a greedy packing coloring and provides a polynomial-time greedy algorithm to obtain packing colorings. It develops a graph-transformation framework via $G(k)$ and the dense maximization procedure (DMP) to characterize $Γ_{ρ}(G)$ in terms independent sets, establishing the fundamental relation $Γ_{ρ}(G)=\max_{A\in\mathcal{I}} DMP(A)$ with an upper bound $Γ_{ρ}(G)\le n-i(G)+1$. The authors derive structural and computational results, including exact characterizations for graphs with large $Γ_{ρ}(G)$, diameter-based formulas ($Γ_{ρ}(G)=n-i(G)+1$ for diam$(G)=2$ and $Γ_{ρ}(G)=n-m(G)+2$ for diam$(G)=3$), and complete analyses for paths and several lattice-related questions. They also compare $χ_{ρ}$ and $Γ_{ρ}$, showing potential large gaps, and discuss open problems and directions for infinite graphs and lattices.

Abstract

A map $c:V(G)\rightarrow\{1,\dots,k\}$ of a graph $G$ is a packing $k$-coloring if every two different vertices of the same color $i\in \{1,\dots,k\}$ are at distance more than $i$. The packing chromatic number $χ_ρ(G)$ of $G$ is the smallest integer $k$ such that there exists a packing $k$-coloring. In this paper we introduce the notion of \textit{Grundy packing chromatic number}, analogous to the Grundy chromatic number of a graph. We first present a polynomial-time algorithm that is based on a greedy approach and gives a packing coloring of $G$. We then define the Grundy packing chromatic number $Γ_ρ(G)$ of a graph $G$ as the maximum value that this algorithm yields in a graph $G$. We present several properties of $Γ_ρ(G)$, provide results on the complexity of the problem as well as bounds and some exact results for $Γ_ρ(G)$.

Figures (5)

• Figure 1: Some packing colorings of selected paths.
• Figure 2: Step-by-step execution of Algorithm 1. The vertex shown in black is colored at each step.
• Figure 3: Graph $P_5(4)$ together with $i(P_5(4))$-set (black vertices).
• Figure 4: An example graph $G$ and its corresponding $G(3)$ together with their $i(G)$- and $i(G(3))$-sets, respectively (black vertices).
• Figure 5: Graph $G$ that fulfills $(iii)$ of Theorem \ref{['n-1']}.

Theorems & Definitions (9)

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