(Georgia Institute of Technology, 2012-05)
Seymour, Paul
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.
(Georgia Institute of Technology, 2012-05)
Seymour, Paul
Let H be a graph and let I be the set of all graphs that do not contain H as a minor. Then H is planar if and only if there is an upper bound on the treewidth of the members of I; and there are many other similar theorems that relate properties of H to structural properties of the members of I. Let us call them "structure theorems". What about structure theorems for other containment relations instead of minor containment? For induced subgraph containment, there are virtually no such theorems known, but for subtournament containment there are some quite pretty structure theorems, mostly excluding tournaments that are almost transitive. A transitive tournament is the tournament analogue of both a stable set and a clique, so what happens if we exclude as induced subgraphs TWO graphs instead of one; one almost a stable set, and the other almost a clique? This turns out to be a better problem with some nice answers. Partly joint work with Maria Chudnovsky, Sasha Fradkin, Ilhee Kim, Gaku Liu, Sergei Norin, Bruce Reed.