2-dimensional unit vector flows
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912926015684608 |
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| author | Houdrouge, Hussein Miraftab, Bobby Morin, Pat |
| author_facet | Houdrouge, Hussein Miraftab, Bobby Morin, Pat |
| contents | We study $2$-dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in $\mathbb R^{3}$. We give a new geometric characterization of $\mathbb S^{2}$-flows on cubic graphs. We also prove that the class of cubic graphs admitting an $\mathbb S^{2}$-flow is closed under a natural composition operation, which yields further constructions; in particular, blowing up a vertex into a triangle preserves the existence of an $\mathbb S^{2}$-flow. Our second contribution is algebraic: we extend the rank-based approach of [SIAM J. Discrete Math., 29 (2015), pp.~2166--2178] from $\mathbb S^{1}$-flows to $\mathbb S^{2}$-flows. More precisely, we show that if an $\mathbb S^{2}$-flow $φ$ satisfies $\operatorname{rank}(S_{\mathbb{Q}}(φ))\le 2$ and $S_{\mathbb{Q}}(φ)$ is odd-coordinate-free, then the graph admits a nowhere-zero $4$-flow. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_21526 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 2-dimensional unit vector flows Houdrouge, Hussein Miraftab, Bobby Morin, Pat Combinatorics Discrete Mathematics 05C21, 15A03 We study $2$-dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in $\mathbb R^{3}$. We give a new geometric characterization of $\mathbb S^{2}$-flows on cubic graphs. We also prove that the class of cubic graphs admitting an $\mathbb S^{2}$-flow is closed under a natural composition operation, which yields further constructions; in particular, blowing up a vertex into a triangle preserves the existence of an $\mathbb S^{2}$-flow. Our second contribution is algebraic: we extend the rank-based approach of [SIAM J. Discrete Math., 29 (2015), pp.~2166--2178] from $\mathbb S^{1}$-flows to $\mathbb S^{2}$-flows. More precisely, we show that if an $\mathbb S^{2}$-flow $φ$ satisfies $\operatorname{rank}(S_{\mathbb{Q}}(φ))\le 2$ and $S_{\mathbb{Q}}(φ)$ is odd-coordinate-free, then the graph admits a nowhere-zero $4$-flow. |
| title | 2-dimensional unit vector flows |
| topic | Combinatorics Discrete Mathematics 05C21, 15A03 |
| url | https://arxiv.org/abs/2602.21526 |