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Reduction Operations and Characterizations of S1-Flows in Graphs

  • Chenxing Li
  • Jiaao Li
  • Rong Luo
  • Bo Su
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Abstract

Thomassen (J. Combin. Theory Ser. B 108 (2014), 81-91) showed that every graph admitting a nowhere-zero $3$-flow also admits an S1-flow. He also proved that the converse holds for cubic graphs, but constructed counterexamples showing it fails in general. Wang et al. (SIAM J. Discrete Math. 29 (2015), 2166-2178) presented a couple of sufficient conditions under which the existence of an S1-flow guarantees the existence of a nowhere-zero 3-flow. In this paper, we first prove that a graph with maximum degree at most four admits a nowhere-zero $3$-flow if and only if it admits an S1-flow. We then develop some reduction techniques for S1-flows based on graph operations including bull-growth, $2$-sums, and contractions. Finally, we apply those techniques to characterize triangularly connected graphs and graphs containing a spanning triangle-tree that admit S1-flows, respectively.

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Cite

BibTeXRIS
BibTeX
@misc{li2026reduction,
  title = {{Reduction Operations and Characterizations of $S^1$-Flows in Graphs}},
  author = {Chenxing Li and Jiaao Li and Rong Luo and Bo Su},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.18725},
  eprint = {2608.18725},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.18725},
}
RIS
TY  - UNPB
TI  - Reduction Operations and Characterizations of S^1-Flows in Graphs
AU  - Chenxing Li
AU  - Jiaao Li
AU  - Rong Luo
AU  - Bo Su
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2608.18725
UR  - https://arxiv.org/abs/2608.18725
AB  - Thomassen (J. Combin. Theory Ser. B 108 (2014), 81-91) showed that every graph admitting a nowhere-zero $3$-flow also admits an S^1-flow. He also proved that the converse holds for cubic graphs, but constructed counterexamples showing it fails in general. Wang et al. (SIAM J. Discrete Math. 29 (2015), 2166-2178) presented a couple of sufficient conditions under which the existence of an S^1-flow guarantees the existence of a nowhere-zero 3-flow. In this paper, we first prove that a graph with maximum degree at most four admits a nowhere-zero $3$-flow if and only if it admits an S^1-flow. We then develop some reduction techniques for S^1-flows based on graph operations including bull-growth, $2$-sums, and contractions. Finally, we apply those techniques to characterize triangularly connected graphs and graphs containing a spanning triangle-tree that admit S^1-flows, respectively.
ER  -