Known proofs of the Four-Color Theorem consist of two steps - reducibility and
discharging. While the discharging part is essentially human-checkable, the
reducibility part is far from it. Reducibility is a useful technique that
allows one to deduce that certain configurations cannot appear in a minimal
counterexample, but it lacks a general theory.

Related notions of reducibility have been developed for the cycle double cover
conjecture of Seymour and Szekeres and the 5-flow conjecture of Tutte. To
settle these conjectures completely using this technique one would need the
reducibility of an infinite sequence of configurations, and therefore at least
a modest theory of reducibility.

We explain the technique of reducibility and discuss new results pertaining to
the Four-Color Theorem and the 5-flow conjecture. This talk is based on joint
work with Robin Thomas.