[Paper Review] Planar graphs with maximum degree D at least 8 are (D+1)-edge-choosable

Marthe Bonamy|arXiv (Cornell University)|Mar 17, 2013

Advanced Graph Theory Research6 references3 citations

TL;DR

This paper proves that every planar graph with maximum degree Δ ≥ 8 is (Δ+1)-edge-choosable, resolving a long-standing conjecture by Borodin for Δ = 8. Using a discharging method with tailored configurations and recoloring arguments in directed graphs, the authors show that no minimal counterexample can exist, thereby confirming Vizing's list coloring conjecture for this class of graphs.

ABSTRACT

We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is k-edge-choosable if for any assignment of k colors to every edge, there is an edge coloring such that the color of every edge belongs to its color assignment. Vizing conjectured in 1965 that every graph is (D+1)-edge-choosable, where D is the maximum degree. In 1990, Borodin solved the conjecture for planar graphs with maximum degree at least 9, and asked whether the bound could be lowered to 8. We prove here that planar graphs with maximum degree D at least 8 are (D+1)-edge-choosable.

Motivation & Objective

• To resolve Borodin's open problem on whether planar graphs with Δ = 8 are (Δ+1)-edge-choosable.
• To close the gap between known results for Δ ≥ 9 and Δ ≤ 7 in the context of Vizing's list edge-coloring conjecture for planar graphs.
• To establish that the (Δ+1)-edge-choosability bound holds for all planar graphs with Δ ≥ 8, extending prior results.

Proposed method

• Employing a discharging method on a hypothetical minimal counterexample graph G with Δ(G) ≤ 8 and χ′_ℓ(G) > 9.
• Defining and forbidding specific configurations (C₁ to C₁₁) through structural analysis and edge list coloring extension arguments.
• Assigning initial weights to vertices (d(v) − 6) and faces (2d(f) − 6) to apply Euler's formula and derive a contradiction.
• Applying a set of discharging rules (R₃ to R₈) that redistribute weights based on neighbor types (weak, semi-weak, E₂/E₃/E₄-neighbors) to ensure non-negative final weights.
• Using directed graph recoloring arguments in key claims (e.g., Claims 3, 4, 6) to extend L-edge-colorings from subgraphs to the full graph.
• Proving that after discharging, the total weight remains non-negative, contradicting Euler’s formula and thus eliminating the existence of a minimal counterexample.

Experimental results

Research questions

• RQ1 Are all planar graphs with maximum degree Δ = 8 (Δ+1)-edge-choosable?
• RQ2 Can the bound Δ ≥ 9 in Borodin’s result on (Δ+1)-edge-choosability be reduced to Δ ≥ 8?
• RQ3 Does the list edge-coloring conjecture hold for planar graphs with Δ ≥ 8, specifically whether χ′_ℓ(G) ≤ Δ(G) + 1?

Key findings

• Every planar graph with Δ ≥ 8 is (Δ+1)-edge-choosable, confirming Conjecture 2 for this case.
• The result resolves Problem 5.9 from Borodin’s 2013 survey on list edge-coloring of planar graphs.
• The proof establishes that no minimal counterexample exists under the given constraints, using a discharging method with forbidden configurations.
• For planar graphs with 5 ≤ Δ ≤ 7, the result implies 9-edge-choosability, a previously unknown bound.
• The method successfully handles configurations that evade standard discharging techniques through directed recoloring arguments.
• The approach is adaptable to prove (Δ+1)-edge-choosability for Δ ≥ 8, though the result is less impactful than the simpler proof for Δ ≥ 9.