[Paper Review] List-coloring embedded graphs without cycles of lengths 4 to 8

Zdenĕk Dvořák, Luke Postle|arXiv (Cornell University)|Aug 14, 2015

Computational Geometry and Mesh Generation15 references16 citations

TL;DR

This paper proves that every planar graph without cycles of lengths 4 to 8 is 3-choosable, resolving a question posed by Borodin. By introducing a novel variant called correspondence coloring—generalizing list coloring—it enables structural reductions previously only valid for ordinary coloring, thereby establishing 3-choosability under the cycle-length restriction.

ABSTRACT

The well-known Steinberg's conjecture postulates that every planar graph without 4-cycles and 5-cycles is 3-colorable. The list-coloring version of this claim is known to be false. However, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin. For the proof, we use a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring.

Motivation & Objective

• To resolve Borodin's question regarding the 3-choosability of planar graphs excluding cycles of lengths 4 to 8.
• To establish that the absence of cycles of length 4 through 8 suffices for 3-choosability, despite the failure of the list-coloring version of Steinberg's conjecture.
• To develop and apply a new coloring framework—correspondence coloring—that generalizes list coloring and enables structural reductions.

Proposed method

• Introduce correspondence coloring as a generalization of list coloring, allowing for more flexible edge-dependent color assignments.
• Use correspondence coloring to simulate structural reductions typically valid only in ordinary coloring, extending their applicability to list-coloring settings.
• Apply discharging arguments within the correspondence coloring framework to analyze the structure of minimal counterexamples.
• Prove that graphs without cycles of length 4 to 8 cannot contain certain reducible configurations under correspondence coloring, leading to a contradiction if 3-choosability fails.
• Leverage the absence of short cycles (4–8) to control the local structure and enforce reducibility conditions in the discharging process.
• Establish that all possible configurations in such graphs are reducible under correspondence coloring, implying 3-choosability.

Experimental results

Research questions

• RQ1 Is every planar graph without cycles of length 4 to 8 3-choosable, despite the failure of the list-coloring version of Steinberg's conjecture?
• RQ2 Can correspondence coloring be used to extend structural reductions from ordinary coloring to list-coloring problems?
• RQ3 Does the exclusion of cycles of length 4 to 8 suffice to guarantee 3-choosability in planar graphs?
• RQ4 What structural properties of planar graphs without cycles of length 4 to 8 make them 3-choosable under correspondence coloring?
• RQ5 How does correspondence coloring enable the proof of 3-choosability where traditional list coloring fails?

Key findings

• Planar graphs without cycles of length 4 to 8 are 3-choosable, confirming Borodin's conjecture in this restricted case.
• The absence of cycles of length 4 to 8 ensures that all potential minimal counterexamples to 3-choosability are structurally reducible under correspondence coloring.
• Correspondence coloring enables the transfer of structural reduction techniques from ordinary coloring to list-coloring, overcoming limitations of standard list coloring.
• The proof establishes that no planar graph without cycles of length 4 to 8 can be a minimal counterexample to 3-choosability under the correspondence coloring framework.
• The result provides a positive answer to Borodin’s question about 3-choosability under cycle-length restrictions, even though the list-coloring version of Steinberg’s conjecture is false.