On the Ban-Linial Conjecture

TL;DR

This paper investigates the Ban-Linial conjecture on external partitions in cubic graphs, proving it holds in two special cases involving cycle-tree decompositions and bipartite complements of cubic trees.

Contribution

The paper proves the Ban-Linial conjecture for specific classes of cubic graphs, advancing understanding of external partitions with balanced size differences.

Findings

Proved the conjecture for graphs decomposable into a cycle and a tree.

Established the conjecture for graphs with a cubic tree whose complement is bipartite.

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.