Polynomial-time decidability of planar 3-edge-colorability

Determine whether there exists a polynomial-time algorithm that, given a planar graph G, decides whether G is 3-edge-colorable (i.e., whether the chromatic index χ′(G) is at most 3).

Background

The paper’s motivation is the longstanding computational status of deciding 3-edge-colorability for planar graphs. For connected 3-regular planar graphs, 3-edge-colorability coincides with 2-connectivity by Tait’s reformulation of the Four-Color Theorem, and can be checked in linear time. However, for general subcubic planar graphs, the problem does not reduce so directly.

The authors paper a related augmentation framework: recognizing when a subcubic planar graph is a subgraph of a k-connected 3-regular planar graph. They resolve all cases for k ≤ 2 in polynomial time and show NP-completeness for k = 3 in the variable-embedding setting, but emphasize that these results do not settle the complexity of planar 3-edge-colorability.