[Paper Review] Graphs with maximum degree D at least 17 and maximum average degree less than 3 are list 2-distance (D+2)-colorable

Marthe Bonamy, Benjamin Lévêque+1 more|arXiv (Cornell University)|Jan 29, 2013

Advanced Graph Theory Research11 references3 citations

TL;DR

This paper proves that graphs with maximum degree Δ ≥ 17 and maximum average degree less than 3 are list 2-distance (Δ+2)-colorable, improving prior results for planar graphs of girth at least 6. The proof uses a novel discharging method based on Brooks' lemma to establish structural properties, enabling a global cactus-like structure that ensures colorability with Δ+2 colors under list constraints.

ABSTRACT

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. It is already known that planar graphs of girth at least 6 and of maximum degree D are list 2-distance (D+2)-colorable when D>=24 (Borodin and Ivanova (2009)) and 2-distance (D+2)-colorable when D>=18 (Borodin and Ivanova (2009)). We prove here that D>=17 suffices in both cases. More generally, we show that graphs with maximum average degree less than 3 and D>=17 are list 2-distance (D+2)-colorable. The proof can be transposed to list injective (D+1)-coloring.

Motivation & Objective

• To determine the minimum maximum degree Δ that ensures list 2-distance (Δ+2)-colorability for sparse graphs.
• To extend known results on 2-distance coloring of planar graphs with girth ≥6 to a broader class of graphs defined by maximum average degree (mad) < 3.
• To improve the threshold from Δ≥18 or Δ≥24 to Δ≥17 for list 2-distance (Δ+2)-coloring in sparse graphs.
• To demonstrate that Brooks’ lemma can be used effectively in global discharging arguments for graph coloring problems.

Proposed method

• A discharging method is applied to graphs with mad(G) < 3 and Δ(G) ≥ 17, assigning initial weights based on vertex degrees.
• Discharging rules (R1.i, R2, R3, R4, Rg) redistribute weights to show that every vertex ends with final weight at least 3, implying mad(G) ≥ 3.
• The contradiction is derived by assuming a minimal counterexample G with Δ(G) ≤ k and no list 2-distance (k+2)-coloring, leading to mad(G) ≥ 3.
• Brooks’ lemma is used to establish a global cactus-like structure in the minimal counterexample, enabling stronger structural constraints.
• The proof is adapted to show that list injective (Δ+1)-coloring is also possible under the same conditions.
• The argument is transposed from planar graphs to the more general class of graphs with mad(G) < 3, generalizing prior results.

Experimental results

Research questions

• RQ1 Can the threshold Δ ≥ 17 be sufficient for list 2-distance (Δ+2)-coloring in graphs with mad(G) < 3?
• RQ2 Does the use of Brooks’ lemma in a discharging framework enable stronger structural conclusions for 2-distance coloring?
• RQ3 Is the list 2-distance (Δ+2)-colorability result for planar graphs of girth ≥6 extendable to graphs with mad(G) < 3?
• RQ4 What is the maximum Δ for which a graph with mad(G) < 3 is not (Δ+2)-list 2-distance colorable?
• RQ5 Can the same method be adapted to prove list injective (Δ+1)-coloring under the same conditions?

Key findings

• Graphs with maximum degree Δ ≥ 17 and maximum average degree less than 3 are list 2-distance (Δ+2)-colorable.
• The threshold Δ ≥ 17 improves upon prior results that required Δ ≥ 18 or Δ ≥ 24 for similar classes of graphs.
• The proof establishes that the minimal counterexample must have mad(G) ≥ 3, contradicting the assumption mad(G) < 3.
• The use of Brooks’ lemma enables a global structural analysis, revealing a cactus-like arborescent structure in the minimal counterexample.
• The method can be directly adapted to show that such graphs are also list injective (Δ+1)-colorable.
• The result generalizes known 2-distance coloring results for planar graphs of girth at least 6 to a broader family of sparse graphs defined by mad(G) < 3.