(1,1,2,2)-Packing Coloring

• (1,1,2,2)-packing coloring is a constrained graph coloring that partitions vertices into two independent sets and two 2-packings with strict distance requirements.
• Key structural insights involve decomposing graphs into square graphs and bipartite substructures, enabling effective coloring of subcubic and claw-free cubic graphs.
• Recent advances use combinatorial proofs and discharging arguments to provide both theoretical bounds and polynomial-time algorithms in challenging graph classes.

A $(1,1,2,2)$-packing coloring is a constrained coloring of a graph where the vertex set is partitioned into four parts—two independent sets (1-packings) and two 2-packings—such that vertices assigned the same color in each part are appropriately separated according to precise distance constraints. This notion generalizes both graph coloring and distance-packing substructures, and plays a critical role in the study of the packing chromatic number, particularly for subcubic and cubic graphs. Recent research has established foundational results and revealed nuanced exceptions and connections to global graph parameters.

1. Formal Definition and Characterizing Frameworks

Let $G = (V, E)$ be a finite simple graph. A subset $P \subseteq V$ is a $k$-packing if any two distinct vertices $u, v \in P$ satisfy $d_G(u, v) > k$, where $d_G(u, v)$ is the length of the shortest path between $u$ and $v$ in $G$. Given a non-decreasing sequence $S=(a_1,a_2,\dots,a_r)$ of positive integers, an $S$-packing coloring is a function $f: V \to \{1, 2, ..., r\}$ such that for every pair $u \neq v$ with $f(u) = f(v) = i$, we have $d_G(u, v) > a_i$.

For $S = (1,1,2,2)$, a $(1,1,2,2)$-packing coloring of $G$ is a partition $V = P_1 \cup P_2 \cup P_3 \cup P_4$, where $P_1, P_2$ are 1-packings (independent sets), and $P_3, P_4$ are 2-packings (the distance between any two in the same $P_3$ or $P_4$ is at least 3) (Mortada et al., 30 Dec 2025, Zein et al., 26 Mar 2025, Brešar et al., 2024, Liu et al., 2019, Brešar et al., 2016).

2. Structural Insights and Key Theoretical Results

A central structural insight is that graphs admitting a $(1,1,2,2)$-packing coloring allow for strong decompositions into highly separated sets. An equivalent condition, originally observed for subcubic graphs, is that $G$ has a partition $V = V_1 \cup V_2 \cup V_3$ where $V_2, V_3$ are independent, and the square graph $G^2[V_1]$ is bipartite. This enables further splitting of the bipartition of $G^2[V_1]$ into the two 2-packings, thus transferring the problem to structural and coloring properties in $G$ and its square (Brešar et al., 2016).

For cubic (degree 3 everywhere) and subcubic graphs ($\Delta(G) \leq 3$), several deep results hold:

• Every non-regular subcubic graph is $(1,1,2,2)$-packing colorable (Zein et al., 26 Mar 2025).
• Every claw-free cubic graph is $(1,1,2,2)$-packing colorable (Mortada et al., 30 Dec 2025, Brešar et al., 2024).
• The Petersen graph is the canonical obstruction among cubic graphs, failing to admit such a coloring, but does admit a $(1,1,2,2,3)$-packing coloring (Brešar et al., 2016).

3. Proof Methodologies and Combinatorial Constructions

Recent advances have produced simplified and combinatorial proofs, particularly for claw-free cubic graphs:

• Triangle Elimination via 2-Packings: Key lemmas establish that in any cubic graph, there exist two disjoint 2-packings whose removal leaves the graph triangle-free. This relies on carefully defined potential functions and exchange arguments maximizing the coverage of triangle-rich vertices (Mortada et al., 30 Dec 2025).
• Odd-Cycle Elimination: After removing triangles, the structure supports the further elimination of odd cycles by augmenting the 2-packings, thereby forcing the bipartiteness of the residual graph. This process relies on claw-freeness, as any contradictory cycle would introduce a forbidden claw (Mortada et al., 30 Dec 2025).
• Split-Extend Approach for General Claw-Free Cubic Graphs: In alternative constructive proofs, claw-free cubic graphs are decomposed via bridge components and then locally extended with canonical colorings on each piece employing perfect matching and 2-factor theorems for the underlying multigraphs (Brešar et al., 2024).

For non-regular subcubic graphs, maximal bipartite induced subgraphs, specialized vertex weightings, and arguments on the bipartiteness of induced squares are utilized, producing broad coverage but not directly yielding polynomial algorithms (Zein et al., 26 Mar 2025).

4. Classes with Guaranteed $(1,1,2,2)$-Packing Colorability

The table below summarizes key graph classes and their status:

Class	$(1,1,2,2)$-Packing Colorable?	Reference
Non-regular subcubic graphs	Yes	(Zein et al., 26 Mar 2025)
Claw-free cubic graphs	Yes	(Mortada et al., 30 Dec 2025, Brešar et al., 2024)
Subcubic graphs with $\operatorname{mad}(G) < 30/11$	Yes	(Liu et al., 2019)
Generalized prisms of cycles (except Petersen)	Yes	(Brešar et al., 2016)
Petersen graph	No (needs 5 colors)	(Brešar et al., 2016)

Further, for planar subcubic graphs of girth at least 8, and subcubic graphs of bounded maximum average degree, $(1,1,2,2)$-packing colorability is guaranteed via discharging arguments (Liu et al., 2019).

5. Connection to Packing Chromatic Number and Open Problems

If a graph $G$ admits a $(1,1,2,2)$-packing coloring, its subdivision $S(G)$ admits a $(1,2,3,4,5)$-packing coloring, placing an upper bound of 5 on the packing chromatic number $\chi_\rho(S(G))$ for these families. This connection is of particular importance for the study of the packing chromatic number for large graph families. The conjecture that all subcubic graphs (except the Petersen graph) satisfy $\chi_\rho(S(G)) \le 5$ remains open, though established for broad subclasses (Brešar et al., 2024, Liu et al., 2019, Brešar et al., 2016).

The only known cubic obstruction is the Petersen graph, which does not permit a $(1,1,2,2)$-packing coloring but does permit a $(1,1,2,2,3)$-packing coloring. The complete characterization of cubic graphs requiring more than four colors is not settled, prompting continuing interest in the limits of the theory (Zein et al., 26 Mar 2025).

6. Algorithmic and Constructive Aspects

Some proofs yield polynomial-time coloring algorithms (notably for claw-free cubic graphs (Brešar et al., 2024)), while others rely on minimal-counterexample and nonconstructive arguments (notably for non-regular subcubic graphs (Zein et al., 26 Mar 2025)). In the constructive approach, recursive or component-based coloring, perfect matching, and 2-factor tools are employed for explicit assignment of color classes.

Discharging arguments enable extension to classes defined by average degree constraints or planarity/girth, yielding broad yet partly nonconstructive existence results (Liu et al., 2019).

7. Further Directions and Open Questions

Current research focuses on several directions:

• Characterizing all cubic graphs that force 5 or more colors in this scheme.
• Extending existence results to broader classes (e.g., relaxing claw-freeness, exploring graphs where every vertex lies on a triangle or 4-cycle).
• Strengthening coloring constraints, e.g., obtaining $(1,1,2,3)$-packing colorings for certain classes.
• Developing efficient algorithms for coloring or recognizing $(1,1,2,2)$-packing colorable graphs in general (Mortada et al., 30 Dec 2025, Brešar et al., 2024).

The methodology and decomposition techniques in recent proofs suggest further exploration may be possible by targeting local forbidden induced subgraphs or average degree constraints, and by leveraging bridging, matching, and 2-factor theory adapted to broader host families.