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I asked this on StackExchange TCS but did not get a satisfactory answer:

Four Color Theorem is equivalent to "Every cubic planar bridgeless graphs is 3-edge colorable". However, 3-edge coloring of cubic graphs is $NP$-complete. I'm wondering if the problem is still hard for cubic planar graphs.

What is the complexity of 3-edge coloring for cubic planar graphs?

Also, It is conjectured that $\Delta$-edge coloring is $NP$-hard for planar graphs with maximum degree $\Delta \in${4,5}.

Has any progress been made towards resolving this conjecture?

Marek Chrobak and Takao Nishizeki. Improved edge-coloring algorithms for planar graphs. Journal of Algorithms, 11:102-116, 1990

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I believe the answer to the first question is "P": see my comment on… –  Colin McQuillan Feb 22 '11 at 19:58

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