Characterization of strongly $\mathbb{Z}_\ell$-connected graphs of small order
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915885666533376 |
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| author | Li, Jiaao Su, Bo Wang, Zhouningxin Wei, Chunyan |
| author_facet | Li, Jiaao Su, Bo Wang, Zhouningxin Wei, Chunyan |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21591 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterization of strongly $\mathbb{Z}_\ell$-connected graphs of small order Li, Jiaao Su, Bo Wang, Zhouningxin Wei, Chunyan Combinatorics 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. |
| title | Characterization of strongly $\mathbb{Z}_\ell$-connected graphs of small order |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.21591 |