On the Ban-Linial Conjecture

Let $G$ be a graph and let $\{X_0,X_1\}$ be a partition of $V(G)$. This partition is called external or unfriendly if every $x\in X_i$ has at least as many neighbours in $X_{1-i}$ as in $X_i$. Every maximum edge-cut gives rise to an external partition, so these partitions are always guaranteed to exist. However, it remains a challenge to find such partitions with additional restrictions. Ban and Linial have conjectured that in the case when $G$ is cubic, there always exists an external partition $\{X_0,X_1\}$ for which $-2\le \mathrm{\mid }{X}_0\mathrm{\mid }-\mathrm{\mid }{X}_1\mathrm{\mid }\le 2$. We prove this in two special cases: whenever $G$ can be decomposed into a cycle and a tree, and whenever $G$ has a cubic tree $T$ for which $G-E(T)$ is bipartite.