Packing six T-joins in plane graphs

Let G be a plane graph and T an even subset of its vertices. It has been conjectured that if all T-cuts of G have the same parity and the size of every T-cut is at least k, then G contains k edge-disjoint T-joins. The case k = 3 is equivalent to the Four Color Theorem, and the cases k = 4 , which was conjectured by Seymour, and k = 5 were proved by Guenin. We settle the next open case k = 6 .

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