2-Distance Coloring of Planar Graphs

• 2-distance coloring of planar graphs is a vertex assignment ensuring distinct colors for vertices within two hops, equivalent to properly coloring the square of the graph.
• Recent research employs discharging methods and combinatorial arguments to tighten chromatic bounds under conditions like high girth, bounded degree, and sparsity.
• Developed techniques such as list coloring and recoloring support practical applications in frequency assignment and network scheduling.

A 2-distance coloring of a planar graph is an assignment of colors to the vertices such that every pair of vertices either adjacent or having a common neighbor receive distinct colors; equivalently, it is a proper coloring of the square of the graph, $G^2$, where vertices at distance at most two in $G$ become adjacent in $G^2$. The paper of 2-distance coloring examines the minimum number of colors necessary, often focusing on additional constraints such as list colorability, bounds on maximum degree $\Delta$, bounds on maximum average degree $\operatorname{mad}(G)$, and girth conditions (the length of the shortest cycle). A central theme in the area is determining quantitative bounds for the 2-distance chromatic number on various classes of planar graphs and understanding the combinatorial structure that enables optimal or near-optimal colorings.

1. Fundamental Definitions and Equivalences

Let $G = (V,E)$ be a planar graph. A 2-distance $k$-coloring is a vertex coloring $f : V \to \{1,\ldots,k\}$ such that $f(u) \ne f(v)$ whenever $u$ and $v$ are at distance at most $2$ in $G$.

• Square of a Graph: $G^2$ is defined on $V(G)$ so that $uv \in E(G^2)$ if and only if $d_G(u,v) \leq 2$.
• List Coloring: Each vertex $v$ is assigned a list $L(v)$ of allowed colors and a coloring is valid if each vertex receives a color from its list and the 2-distance condition is satisfied.

The minimal number $k$ for which $G^2$ is (list-)colorable is called the (list) 2-distance chromatic number, denoted $\chi_2(G)$ or $\chi^2_\ell(G)$.

The natural lower bound is $\Delta + 1$, as every vertex and its $\Delta$ neighbors must all receive distinct colors. The paper aims to establish whether $\Delta+1$ colors suffice and to determine the minimum possible additive constant for broader classes of planar graphs.

2. Key Theorems and Historical Progress

Recent advances have improved upper bounds on $\chi_2(G)$, often under degree and sparsity constraints:

Paper/Authors	Conditions	Bound	arXiv id
Borodin & Ivanova (2009), Zhu & Bu	$\Delta \geq 18$, girth $\geq 6$	$\chi_2(G) \leq \Delta+2$	-
(Bonamy et al., 2013)	$\Delta \geq 17$, $\operatorname{mad}(G)<3$	List $\chi_2(G) \leq \Delta+2$	(Bonamy et al., 2013)
(Deniz, 2022)	$\Delta\geq 6$, girth $\geq 6$	$\chi_2(G)\leq \Delta+4$	(Deniz, 2022)
(La et al., 2021)	$\Delta\geq 7$, $\operatorname{mad}(G)<18/7$	$\chi_2(G)\leq \Delta+1$	(La et al., 2021)
(La et al., 2021)	$\Delta\geq 6$, $\operatorname{mad}(G)<8/3$	$\chi_2(G)\leq \Delta+2$	(La et al., 2021)
(Deniz, 2023, Aoki, 2023, Aoki, 25 Jun 2024)	Planar, $\Delta\leq5$, $\Delta\leq6$	$\chi_2(G)\leq 16,17,20$	(Deniz, 2023, Aoki, 2023, Aoki, 25 Jun 2024)
(Deniz, 18 Mar 2024, Bousquet et al., 2021)	Planar, arbitrary $\Delta$	$\chi_2(G)\leq 2\Delta+7$	(Deniz, 18 Mar 2024)

These bounds have typically been achieved via intricate combinatorial arguments using minimal counterexamples, structural lemmas, and the discharging method.

3. Discharging Methodology and Structural Lemmas

The discharging method is central to almost all recent results. The process involves:

• Assigning "charge" to each vertex ($d(v)-c$, with $c$ an appropriate constant) and face ($l(f)-c$) based on the degree and face length.
• Global conservation of total charge (negative due to Euler's formula for planar graphs).
• Defining local transfer rules (discharging rules) based on degrees, face sizes, and configurations of vertices ("weak," "semiweak," "strong," "support," etc).
• Proving that after redistributing charge, every element has nonnegative final charge, contradicting the initial total.

For example, in (Bonamy et al., 2013), the initial weight is $d(v)$; the rules propagate fractions to control small degree vertices and forbidden local structures which are then shown to be reducible if present. The proof relies on a minimal counterexample approach, forbidding configurations (C₁–C₁₁), and finishing with a discharging argument that ensures $\operatorname{mad}(G)\geq3$, contradicting hypotheses for planar graphs of girth at least $6$.

Additional techniques include recoloring arguments, using the potential method (as in (La et al., 2021)), or auxiliary constructions such as support vertex graphs.

4. Implications for Planar Graphs and Sparsity

The strongest results apply to planar graphs with high girth or bounded maximum average degree, which implies greater sparsity.

• Girth Constraints: Planar graphs with girth $g$ satisfy $(\operatorname{mad}(G)-2)(g-2)<4$ by Euler's formula. Girth $g\geq6$ enforces $\operatorname{mad}(G)<3$, enabling sharper coloring bounds.
• Large Maximum Degree Regime: List 2-distance $(\Delta+2)$-colorability is proven for planar graphs of girth 6 with $\Delta\geq17$ (Bonamy et al., 2013) and further improvements are made for larger girth in more recent work.
• Smaller Degree and Girth Regimes: For planar graphs with $\Delta\leq6$, $g\geq6$, bounds of $20$ or better are achieved (Aoki, 25 Jun 2024). For $g\geq10$, results refine to list 2-distance $(\Delta+2)$ (La et al., 2021).

These bounds approach optimality in light of lower bounds from vertex neighborhoods and known extremal constructions.

5. Connections to Other Coloring Variants

• Injective Coloring: Requires only that colors differ between vertices sharing a neighbor. The list injective $(\Delta+1)$-coloring bound is closely related to 2-distance coloring, often proven via similar methods (Bonamy et al., 2013).
• Exact Square Coloring: Distinct colors between vertices at distance exactly $2$. For triangle-free graphs, this coincides with injective coloring (Foucaud et al., 2020).
• Strong Edge Coloring: Coloring edges so that edges at distance $\leq2$ differ; equivalent to coloring the square of the line graph, with strong results for cubic and high-girth planar graphs (Hudák et al., 2013).

6. Open Problems and Future Directions

Important directions posed in (Bonamy et al., 2013) and subsequent literature include:

• Determining the largest possible $\Delta$ for which there exists a graph with $\operatorname{mad}(G)<3$ not admitting a 2-distance $(\Delta+2)$-coloring. The paper shows $\Delta\leq16$ must be the limit if not tight.
• Lowering the minimum degree constraints, refining bounds for lower girth or different host surfaces.
• Extending proof techniques to additional coloring paradigms (list coloring, strong coloring) or broader graph classes.
• Development of automated discharging and LP-based frameworks, as pioneered in recent work, for pushing bounds further (Bousquet et al., 2022).

7. Practical Applications

2-distance coloring models interference constraints in frequency assignment problems, register allocation, and channel assignment in wireless networks. Improving bounds leads to more efficient allocations under the constraint that "nearby" elements must differ, directly influencing theoretical limits in such applications. In computer science, advances inform coloring and scheduling algorithms for sparse networks or systems modeled by planar graphs.

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This topic remains an active area of graph coloring research, with recent progress moving toward linear or nearly optimal colorings for broader regimes of planar graphs, and with significant interplay between structural combinatorics and algorithmic techniques.