Packing Coloring of Graphs

• Packing coloring is a vertex coloring method where any two vertices with color i must be at a distance greater than i, defining the packing chromatic number.
• The concept generalizes to S-packing colorings and interpolates between standard and distance-2 colorings, with applications in frequency assignment, coding theory, and resource allocation.
• Both bounded and unbounded cases have been studied across various graph families, with NP-completeness results and constructive techniques shaping algorithmic approaches.

A packing coloring of a graph is a vertex coloring in which, for each color $i$, any two vertices assigned color $i$ must be separated by distance greater than $i$ in the graph. The corresponding minimum number of colors needed is the packing chromatic number, a parameter that captures a tradeoff between classical coloring and distance constraints. Packing coloring generalizes to $S$-packing coloring, where a sequence $S=(s_1,s_2,\ldots,s_k)$ specifies the required minimum separations for each color. This subject links issues of extremal combinatorics, graph structure, probabilistic and algorithmic methods, and has connections to frequency assignment, coding theory, and distributed resource allocation.

1. Core Definitions and Variants

Let $G=(V,E)$ be a simple undirected graph. For vertices $u,v$, $d_G(u,v)$ denotes the standard shortest-path distance. A packing $k$-coloring is a function $c:V\to \{1,\ldots,k\}$ such that $c(u)=c(v)=i$ implies $d_G(u,v)>i$. The packing chromatic number $\chi_\rho(G)$ is the smallest $k$ for which such a coloring exists. More generally, for a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, an $S$-packing coloring of $G$ is a partition $V=V_1\dot{\cup}\cdots\dot{\cup}V_k$ such that every two distinct vertices in $V_j$ satisfy $d_G(u,v)>s_j$. The minimum $k$ for which such a coloring exists is the $S$-packing chromatic number of $G$ (Mortada et al., 24 Mar 2025).

A frequent special case is $S=(1,2,\ldots,k)$, which recovers the standard packing coloring. For $(1^a,2^b)$—that is, $a$ colors with distance at least $2$, and $b$ colors with distance at least $3$—one obtains colorings that interpolate between proper and distance-2 (square) colorings (Choi et al., 3 Sep 2025). The notion further admits generalization to $(d,n)$-packing colorings, defined by sequences $s_i = d + \lfloor (i-1)/n \rfloor$ (Deng et al., 2019).

2. Existence and Upper Bounds in Bounded Degree Graphs

The existence and parameters of packing and $S$-packing colorings change dramatically depending on degree constraints, connectivity, local structure (e.g., induced cycles), and when restricted to classes such as planar, outerplanar, or subcubic graphs.

The central broad result for bounded-degree graphs is that every graph of maximum degree $k \geq 3$ admits a $(1^{k-1},2^k)$-packing coloring (Mortada et al., 24 Mar 2025). The proof uses a maximal lex-optimal sequence of $k-1$ independent sets, followed by coloring the remaining vertices through $k$-coloring of the conflict graph in the square $G^2$ (by Brooks’ theorem). As a consequence, all vertices colored with 2-colors are separated by at least distance 3.

For $t$-saturated graphs, sharper results hold. A $k$-degree graph is $t$-saturated if each degree-$k$ vertex is adjacent to at most $t$ other degree-$k$ vertices. Then:

• Every $0$-saturated $k$-degree graph admits a $(1^{k-1},3)$-packing coloring.
• For $1\leq t\leq k-2$, every $t$-saturated $k$-degree graph admits a $(1^{k-1},2)$-packing coloring.
• Every $(k-1)$-saturated $k$-degree graph ($k\geq4$) admits a $(1^{k-1},2^{k-1})$-packing coloring (Mortada et al., 24 Mar 2025).

These results generalize, and often refine, earlier theorems for subcubic graphs, especially when the local degree pairing structure is controlled (Bazzal, 2024).

3. Packing Coloring in Subcubic Planar and Outerplanar Graphs

The packing chromatic number is unbounded for general subcubic graphs (Laïche et al., 2018), but several important subclasses remain bounded:

• Every subcubic planar graph is $(1,2^5)$-packing colorable, and this is sharp—there exists an infinite family of such graphs not $(1,2^4)$-packing colorable (Liu et al., 2024). The proof is via a minimal counterexample and discharging technique, paralleling the methods underlying the Four Color Theorem.
• Every 2-connected bipartite subcubic outerplanar graph has $\chi_\rho(G)\leq 7$. Furthermore, every subcubic triangle-free outerplanar graph admits a $(1,2,2,2)$-packing coloring; however, with a triangle present this may fail. Not every subcubic triangle-free outerplanar graph admits a $(1,2,2,3)$-packing coloring (Brešar et al., 2018).

On the algorithmic and list coloring side, every subcubic graph is both packing $(1^1,2^6)$-choosable and packing $(1^2,2^3)$-choosable, with these bounds sharp due to the Petersen graph (Choi et al., 3 Sep 2025).

4. Packing Coloring in Special Graph Families and Products

The packing chromatic number remains tightly bounded in paths, cycles, coronae, Sierpiński-type graphs, and their products:

• $\chi_\rho(P_n)\leq 3$ and $\chi_\rho(C_n)\leq 4$ for all $n$, with exact values characterized (Furmańczyk et al., 16 Nov 2025).
• The corona $P_n\odot pK_1$ and $C_n\odot pK_1$ have packing chromatic number determined exactly for all $p,n$ (Daouya et al., 2015).
• For path-aligned graph products, constant upper bounds exist regardless of unbounded diameter. For instance, for the product $P_{\ell t}\Diamond_\ell C_n$ (connecting copies of $C_n$ along $P_{\ell t}$), the packing chromatic number is always at most $6$, with conjectured tightness at $5$ except possibly for the $C_5$ case (Furmańczyk et al., 16 Nov 2025).
• For generalized Sierpiński graphs, $\chi_\rho(S_3^n)=8$ for all $n\geq 5$ (improved via explicit construction) (Korze et al., 2018), but is unbounded for $k$-base Sierpiński graphs with $k\geq 4$ (Brešar et al., 2017). For the Sierpiński triangle graphs $ST_3^n$, there is a uniform upper bound of $31$, later brought down to $20$ (Korze et al., 2018).

For Cartesian powers and hypercubes, the packing chromatic number grows $\Theta(2^n)$. Explicit code-based constructions using extended Hamming codes have produced the best asymptotic upper bounds for hypercubes (Gregor et al., 2023).

5. Algorithmic and Combinatorial Complexity

Determining $\chi_\rho(G)$ is NP-complete for $k\geq 4$ even on restricted graph classes (chordal graphs of diameter at least 3, trees) (Kim et al., 2017). It is also hard to approximate within $n^{1/2-\epsilon}$ unless P=NP. For interval graphs of bounded diameter $d$, the problem is fixed-parameter tractable (FPT): one can determine $\chi_\rho(G)$ in $O(n^{d \ln 5d})$ time. The partial coloring (“maximum coverage”) variant is solvable in $O(n^{k+2})$ time for fixed $k$.

For $S$-packing colorings in bounded-degree graphs, the results of (Mortada et al., 24 Mar 2025) and (Bazzal, 2024) are constructive in the sense that they rely on maximal independent set or bipartite spanning subgraph decompositions, but optimal solutions remain NP-hard for general graphs due to the inherent hardness of the independent set problem.

A related parameter is the Grundy packing chromatic number $\Gamma_\rho(G)$, defined as the maximum number of colors used by a greedy first-fit packing coloring algorithm over all vertex orderings. Computing $\Gamma_\rho(G)$ is also NP-complete (Gözüpek et al., 2024).

6. Structural and Extremal Behavior

Packing chromatic numbers exhibit complex and sometimes unintuitive behavior under graph operations:

• Many “critical” graphs (that is, those for which all proper subgraphs have strictly lower $\chi_\rho$) exist and are characterized in various cases (Brešar et al., 2019).
• $\chi_\rho$ is not monotone under edge addition, removal, or usual graph products. In particular, counterexamples exist showing that for some graphs $G,H$,

$$\chi_\rho(G \square H) > \max\{\chi_\rho(G)|H|, \chi_\rho(H)|G|\},$$

showing no universal product bound (Gregor et al., 2023).

Edge subdivision always reduces complexity: while subcubic graphs $G$ have unbounded $\chi_\rho(G)$, their full subdivisions $D(G)$ have $\chi_\rho(D(G))\leq 8$ (Balogh et al., 2018). Subdividing each edge once is thus sufficient to guarantee bounded packing chromatic number regardless of maximum degree, as long as $\Delta(G)\leq 3$.

Tables and explicit classifications are available for many structured classes. For example, the packing chromatic numbers of cubic graph families such as circular ladders, H-graphs, and generalized H-graphs are fully enumerated (Laïche et al., 2018):

Family	$\chi_\rho$
circular ladder $\mathrm{CL}_n$	$5$ (if $n=3$ or $n$ even, $n\notin\{8,14\}$); $6$ otherwise; $7$ for $n\in\{7,8,9\}$
$H(r)$ (even $r$)	$5$
$H(r)$ (odd $r$)	$7$
generalized $H^\ell(r)$ (even $r$)	$5$
generalized $H^\ell(r)$ (odd $r$)	$6$

7. Open Problems and Future Directions

Major open questions include:

• The optimal upper bound for the packing chromatic number of full subdivisions of subcubic graphs. The standing conjecture is that $\chi_\rho(D(G))\leq 5$ for all subcubic $G$ (Balogh et al., 2018, Liu et al., 2019).
• Characterizing families of graphs (e.g., caterpillars, block graphs, generalized Sierpiński graphs) with prescribed or bounded packing chromatic number, and efficient recognition of such graphs (Furmańczyk et al., 16 Nov 2025, Brešar et al., 2017).
• Improvement of upper bounds for the Sierpiński triangle graphs, with best current known $\chi_\rho\leq 20$ (Korze et al., 2018).
• Exploration of parameterized, FPT, or approximation algorithms for packing coloring on families with bounded clique-width or tree-width (Kim et al., 2017).
• Analysis of the gaps between constructive upper bounds (algorithmic, maximal greedy approach) and extremal lower bounds (critical graphs, high-density examples).

Packing coloring remains fundamentally intertwined with both structural graph theory and algorithm design, with further potential extensions to distance $d$-colorings, list variants, and applications in frequency allocation and network resource planning.