The Ban–Linial conjecture for cubic graphs

Let $G=(V,E)$ be a finite simple graph. A split of $G$ is a pair $(X,Y)$ of disjoint sets whose union is $V$. It is external if, after deleting all edges between $X$ and $Y$, every vertex has degree at most half its degree in $G$. A graph is cubic if every vertex has degree $3$.

Ban–Linial conjecture. Every cubic graph has an external split $(X,Y)$ satisfying

$$-2 \leq |X|-|Y| \leq 2.$$

The conjecture asks for an external split that is nearly balanced, strengthening the existence of an external split supplied by any maximum edge-cut. The source proves the claim in two special cases: graphs decomposable into a cycle and a tree, and graphs having a cubic tree $T$ such that $G-E(T)$ is bipartite. The supplied status evidence concerns a stronger conjecture, not this stated Ban–Linial conjecture itself.