Packing Coloring in Graph Theory

• Packing coloring is a graph vertex coloring method where vertices are partitioned into color classes with required minimum distances, generalizing standard colorings.
• Theoretical results, including Brooks-type and saturation theorems, provide insights into optimal color assignments in bounded-degree and subcubic graphs.
• Computational studies reveal NP-completeness in general cases and fixed-parameter tractable algorithms for special graph classes, underlining its network optimization applications.

A packing coloring of a graph specifies a vertex coloring subject to distance-based separation constraints for each color class. Formally, for a given sequence $S=(s_1, s_2, \ldots, s_p)$ of non-decreasing positive integers, an $S$-packing coloring of a graph $G$ is a partition of $V(G)$ into disjoint sets $V_1, \ldots, V_p$ such that any two distinct vertices in $V_i$ are at a distance strictly greater than $s_i$ in $G$. The classical case is the packing chromatic number, where $S$ is the infinite sequence $(1,2,3,\ldots)$ and seeks the minimal $k$ so that the coloring constraint holds for all $1 \leq i \leq k$. This framework generalizes standard colorings, $i$-independent sets, and graph powers, and is central in frequency assignment, network resource allocation, and combinatorial optimization.

1. Formal Definitions and Framework

Given a finite non-decreasing sequence $S=(s_1, s_2, \ldots, s_p)$, an $S$-packing coloring partitions the vertex set of a graph $G$ into $p$ classes $(V_1, \ldots, V_p)$ such that: $$\forall\, 1 \leq i \leq p: \quad \forall\, u, v \in V_i,\, u \neq v:\quad \mathrm{dist}_G(u, v) > s_i$$ This can be equivalently viewed as having each color $i$ define an independent set in the $s_i$-th power $G^{s_i}$. The packing chromatic number, denoted $\chi_\rho(G)$ or $\chi_P(G)$, is the smallest $k$ for which $G$ admits a packing $(1,2,\ldots,k)$-coloring.

For mixed $(1^\ell, 2^k)$-packing colorings, $V(G)$ is partitioned into $\ell$ independent sets (distance at least 2) and $k$ 2-packings (distance at least 3). These interpolate between proper coloring and coloring the graph square.

2. Theoretical Foundations: Brooks-type and Saturation Results

Extending Brooks' theorem to $S$-packing colorings on bounded-degree graphs yields a rich set of results:

• Theorem A (General $k$-degree graphs):

Any graph with $\Delta(G) \leq k$ is $(1^{k-1},2^k)$-packing colorable.

• Theorem B (0-saturated graphs):

For $k \geq 3$, any 0-saturated graph (i.e., no $k$-vertex is adjacent to another $k$-vertex) is $(1^{k-1}, 3)$-packing colorable.

• Theorem C ($t$-saturated graphs, $1 \leq t \leq k-2$):

Any $t$-saturated graph is $(1^{k-1},2)$-packing colorable.

• Theorem D ($(k-1)$-saturated graphs, $k \geq 4$):

Such graphs are $(1^{k-1},2^{k-1})$-packing colorable.

Key techniques include the use of maximum weight independent sets, iterative splitting into independent sets according to prescribed weights, path-structure analysis, and Brooks’ theorem on graph power coloring. These results have recently been generalized to improved bounds for subcubic and planar graphs (Mortada et al., 24 Mar 2025).

Subcubic and Special Cases

For subcubic graphs ($\Delta(G)\leq 3$), refinements using saturation and "heavy" vertices (vertices with all neighbors of maximum degree) yield sharper results:

• Any 1-saturated subcubic graph is $(1,1,3,3)$-packing colorable and $(1,2,2,2,2)$-packing colorable.
• Every $(3,0)$-saturated subcubic graph is $(1,2,2,2,2,2)$-packing colorable (Mortada et al., 10 Jul 2024).
• For planar subcubic graphs, every such graph is $(1,2^5)$-packing colorable, and this bound is tight up to planarity and the Petersen graph exception (Liu et al., 22 Aug 2024).

3. Computational Complexity and Algorithmic Results

The packing coloring problem is generally computationally intractable:

• Determining $\chi_P(G)$ is NP-complete for chordal graphs with diameter at least 3; easy for diameter 2 (maximum independent set suffices) (Kim et al., 2017).
• There is no polynomial-time approximation algorithm achieving an $n^{1/2-\varepsilon}$-factor for the packing chromatic number unless P=NP.
• The problem is fixed-parameter tractable (FPT) in multiple restricted graph classes and parameters (clique-width, treewidth, modular width, maximum clique for unit interval graphs, etc.).
• For interval graphs of bounded diameter $d$, the problem admits an XP algorithm with time $O(n^{d\ln(5d)})$.
• Optimizing the partial packing coloring (maximizing the number of colored vertices with a given color budget) is FPT in $k$ and clique-width (Kim et al., 2017).

Class	Packing Coloring	Partial $k$-coloring
General graphs	NP-c, hard to approx	NP-c (decision); opt open
Chordal diameter 2	P	P
Chordal diameter $\geq3$	NP-c	NP-c
Interval (diam $\leq d$)	XP($d$)	$O(n^{k+2})$ for fixed $k$
Clique-width $q$, fixed $k$	FPT($q,k$)	FPT($q,k$)
Treewidth, fixed $k$	FPT($\text{tw}+k$)	FPT($\text{tw}+k$)

For path-aligned products and caterpillar graphs, packing chromatic numbers remain bounded by small constants independent of graph diameter (Furmańczyk et al., 16 Nov 2025).

4. Structural and Extremal Properties

Packing colorings show strong connections to classical extremal parameters and forbidden subgraphs:

• For all subcubic graphs with maximum average degree $<30/11$, a $(1,1,2,2)$-packing coloring exists. Subdivisions of such graphs have $\chi_p \le 5$ (Liu et al., 2019).
• There is a complete structural characterization of caterpillars with packing chromatic number 3 in terms of spine length modulo 4 and the pattern of leaf attachments (Furmańczyk et al., 16 Nov 2025).
• For path-aligned products, explicit coloring patterns and integer-linear programming have been used to construct and verify optimal bounds. For the family $P_n \Diamond_\ell K_5$, the chromatic number is established as at most 14.

Extremal constructions demonstrate tightness: there exist infinite families of subcubic planar graphs not $(1,2^4)$-packing colorable, and the Petersen graph is a canonical exception in several conjectures (Liu et al., 22 Aug 2024).

5. Applications and Motivations

The packing coloring problem models interference constraints in frequency assignment (where distinct frequencies must be separated according to transmit power) and more generally, resource distribution in networks, smart city infrastructure, and biological diversity management (Furmańczyk et al., 16 Nov 2025). The formalism captures the trade-off between minimizing color (resource) use and maintaining required separation in a spatial or network topology.

The mixed $(1^\ell, 2^k)$-packing framework interpolates between proper coloring and coloring of the graph square, offering a flexible modeling tool for hybrid interference regimes and cross-layer network design (Liu et al., 22 Aug 2024).

6. Open Problems and Future Directions

Recent research highlights several significant directions:

• Does every subcubic graph except the Petersen graph admit a $(1,2^5)$-packing coloring? This is open outside the planar case (Liu et al., 22 Aug 2024).
• For general bounded-degree graphs, does every $k$-degree graph admit a $(1^{k-1},2^{k-1},3)$-packing coloring (Mortada et al., 24 Mar 2025)?
• Is every 0-saturated $k$-degree graph $(1^{k-2},2^k)$-packing colorable?
• Is there a polynomial-time recognition algorithm for caterpillars with $\chi_{\rho}(G)\le4$ (Furmańczyk et al., 16 Nov 2025)?
• Classification of graphs $G$ with $\omega(G)=\chi(G)$ that also satisfy $\chi_\rho(G)=\chi(G)$ remains an open combinatorial question.

A plausible implication is that further exploration of $S$-packing colorings with general sequences and new weighted independent set techniques could resolve several longstanding conjectures and unify distance-based colorings across graph theory and applied network science.

7. References to Key Research

• Mortada & Togni, "On $S$-packing Coloring of Bounded Degree Graphs" (Mortada et al., 24 Mar 2025)
• Mortada & Togni, "Further Results and Questions on $S$-Packing Coloring of Subcubic Graphs" (Mortada et al., 10 Jul 2024)
• Shao et al., "Packing coloring of graphs with long paths" (Furmańczyk et al., 16 Nov 2025)
• Cheng, Lemoine, and Montassier, "Partition subcubic planar graphs into independent sets" (Liu et al., 22 Aug 2024)
• Brešar, Ferme, Kovše, and Togni, "Packing $(1,1,2,2)$-coloring of some subcubic graphs" (Liu et al., 2019)
• Fiala, Kratochvíl, and Golovach, "Notes on complexity of packing coloring" (Kim et al., 2017)

These works collectively define the present landscape and critical results in the paper of the packing coloring problem and its generalizations.