Tutte's Three-Edge-Colouring Conjecture
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The four-colour theorem is equivalent to the statement that every planar cubic graph with no cut-edge is 3-edge-colourable. What about non-planar cubic graphs? The Petersen graph is not 3-edge-colourable, and in 1966 Tutte conjectured that every cubic graph with no cut-edge that does not contain the Petersen graph as a minor is 3-edge-colourable. In 1996 we proved this conjecture, but did not publish the result, for reasons that escape me. We are currently getting it all back together for publication; and this talk is an outline. A graph is "apex'' if deleting some vertex makes it planar; and "doublecross'' if it can be drawn in the plane with crossings, but with only two crossings and both incident with the same region. Apex and doublecross cubic graphs do not have Petersen minors. Joint work with Neil Robertson and Robin Thomas.