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Grötzsch’s conjecture for 2-connected subcubic planar graphs

Prove that every 2-connected planar graph G with maximum degree at most 3 is 3-edge-colorable unless G has exactly one vertex of degree 2.

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Background

The authors note that planar 3-edge-colorability reduces to checking the blocks (maximal 2-connected subgraphs). A simple counting argument shows that a 2-connected subcubic graph with exactly one degree-2 vertex is not 3-edge-colorable.

Grötzsch’s conjecture (as attributed by Seymour) posits that this is the only obstruction in the planar 2-connected subcubic setting. If true, it would imply a polynomial-time algorithm for planar 3-edge-colorability by simple structural checks.

References

Conjecture [Grötzsch, cf.] If~$G$ is a $2$-connected planar graph of maximum degree~$\Delta(G) \leq 3$, then~$G$ is $3$-edge-colorable, unless it has exactly one vertex of degree~$2$.

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