Barnette's conjecture on simple separating non-contractible cycles

A simple separating non-contractible cycle (SNCC) is a simple cycle in a triangulation that is both separating and non-contractible. Let $t$ be an arbitrary triangulation without loops or multiple edges, of genus at least $2$. Barnette's conjecture. The triangulation $t$ contains a simple SNCC.

This is a longstanding deterministic conjecture, despite several advances. It asks whether every such triangulation has a simple separating non-contractible cycle; the paper notes that the conjecture remains open.