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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.18913 |
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Table of Contents:
- 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.