Realizing degree sequences with $\mathcal S_3$-connected graphs
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915172157751296 |
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| author | Guan, Rui Jiang, Chenglin Lai, Hong-Jian Li, Jiaao Li, Xinyuan |
| author_facet | Guan, Rui Jiang, Chenglin Lai, Hong-Jian Li, Jiaao Li, Xinyuan |
| contents | A graph $G$ is $\mathcal S_3$-connected if, for any mapping $β: V (G) \mapsto {\mathbb Z}_3$ with $\sum_{v\in V(G)} β(v)\equiv 0\pmod3$, there exists a strongly connected orientation $D$ satisfying $d^{+}_D(v)-d^{-}_D(v)\equiv β(v)\pmod{3}$ for any $v \in V(G)$. It is known that $\mathcal S_3$-connected graphs are contractible configurations for the property of flow index strictly less than three. In this paper, we provide a complete characterization of graphic sequences that have an $\mathcal{S}_{3}$-connected realization: A graphic sequence $π=(d_1,\, \ldots,\, d_n )$ has an $\mathcal S_3$-connected realization if and only if $\min \{d_1,\, \ldots,\, d_n\} \ge 4$ and $\sum^n_{i=1}d_i \ge 6n - 4$. Consequently, every graphic sequence $π=(d_1,\, \ldots,\, d_n )$ with $\min \{d_1,\, \ldots,\, d_n\} \ge 6$ has a realization $G$ with flow index strictly less than three. This supports a conjecture of Li, Thomassen, Wu and Zhang [European J. Combin., 70 (2018) 164-177] that every $6$-edge-connected graph has flow index strictly less than three. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Realizing degree sequences with $\mathcal S_3$-connected graphs Guan, Rui Jiang, Chenglin Lai, Hong-Jian Li, Jiaao Li, Xinyuan Combinatorics A graph $G$ is $\mathcal S_3$-connected if, for any mapping $β: V (G) \mapsto {\mathbb Z}_3$ with $\sum_{v\in V(G)} β(v)\equiv 0\pmod3$, there exists a strongly connected orientation $D$ satisfying $d^{+}_D(v)-d^{-}_D(v)\equiv β(v)\pmod{3}$ for any $v \in V(G)$. It is known that $\mathcal S_3$-connected graphs are contractible configurations for the property of flow index strictly less than three. In this paper, we provide a complete characterization of graphic sequences that have an $\mathcal{S}_{3}$-connected realization: A graphic sequence $π=(d_1,\, \ldots,\, d_n )$ has an $\mathcal S_3$-connected realization if and only if $\min \{d_1,\, \ldots,\, d_n\} \ge 4$ and $\sum^n_{i=1}d_i \ge 6n - 4$. Consequently, every graphic sequence $π=(d_1,\, \ldots,\, d_n )$ with $\min \{d_1,\, \ldots,\, d_n\} \ge 6$ has a realization $G$ with flow index strictly less than three. This supports a conjecture of Li, Thomassen, Wu and Zhang [European J. Combin., 70 (2018) 164-177] that every $6$-edge-connected graph has flow index strictly less than three. |
| title | Realizing degree sequences with $\mathcal S_3$-connected graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.18100 |