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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.21591 |
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Table of Contents:
- A graph is strongly $\Z_{\ell}$-connected if for each boundary function $β: V(G)\mapsto \Z_{\ell}$ with $β(v) \equiv d(v) \pmod{2}$ for every vertex $v$ and $\sum_{v \in V(G)} β(v) \equiv 0 \pmod{2\ell}$, there exists an orientation $D$ of $G$ such that $d_D^+(v) - d_D^-(v) \equiv β(v) \pmod{2\ell}$ for each $v \in V(G)$. This is a useful notion for studying circular flows of graphs. This note presents a fully self-contained, manual proof of a characterization of $4$-vertex strongly $\mathbb{Z}_\ell$-connected graphs for any integer $\ell\geq 2$, which will be used in our further study in this topic.