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The Δ = 6 case of planar edge-coloring (Vizing’s planar case)

Determine whether every planar graph of maximum degree 6 is 6-edge-colorable.

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Background

Vizing conjectured that all planar graphs with maximum degree Δ ≥ 6 are Δ-edge-colorable. It is now known for Δ ≥ 7, with efficient algorithms available, but the boundary case Δ = 6 remains unresolved.

The authors summarize the current status: while Δ ≥ 7 is settled positively, there exist counterexamples for Δ = 3,4,5 to Δ-edge-colorability, and for Δ = 4,5 the decision problem is suspected to be NP-complete for planar graphs.

References

The case $\Delta = 6$ is still open, while for $\Delta = 3,4,5$ there are planar graphs of maximum degree $\Delta$ that are not $\Delta$-edge-colorable, and at least for $\Delta = 4,5$ the $\Delta$-Edge-Colorability-problem is suspected to be \NP-complete for planar graphs.

— Recognition Complexity of Subgraphs of k-Connected Planar Cubic Graphs (2401.05892 - Goetze et al., 11 Jan 2024) in Section 6 (Discussion)