Tutte seminar - Bruce Richter

Recent Progress on Tutte's 3-flow Conjecture

Abstract: 

Tutte's 3-flow conjecture asserts that every graph with no 1- or 3-edge-cut has an orientation so that, at every vertex, the number of arcs pointing in is the same (modulo 3) as the number of arcs pointing out. It is known that it suffices to prove that every 5-edge-connected graph has such an orientation.

Recently, Thomassen made a breakthrough, proving this for 8-edge-connected graphs. There is now a very simple proof that it holds for 6-edge-connected graphs. Indeed, this is proved by showing that, modulo 3, prescribing the individual out-degrees at every vertex (to sum to |E(G)| modulo 3) is achievable in 6-edge-connected graphs.

With Thomassen and Younger, we proved that this stronger result also holds for 5-edge-connected planar graphs. Both proofs will be outlined in this talk.