On the Ban-Linial Conjecture
Matt DeVos, Kathryn Nurse
TL;DR
The paper investigates the Ban-Linial conjecture, which asks whether every cubic graph has an external split close to a bisection. It introduces and leverages the concepts of splits, discrepancy, and nearly external bisections, connecting these to flows and bisection strategies. The authors prove the conjecture in two structural settings: (i) when the graph can be decomposed into a tree and a cycle, and (ii) when the graph contains a cubic tree T such that G−E(T) is bipartite. Central to the approach are a cubic-tree lemma, reduction rules, and a path from nearly external bisections to the desired external split, offering new constructive methods and linking to Tutte’s 5-flow framework.
Abstract
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 $-2 \le |X_0| - |X_1| \le 2$. 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.