Geometric description of $d$-dimensional flows of a graph
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| Format: | Preprint |
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2025
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| _version_ | 1866912664052039680 |
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| author | Mattiolo, Davide Mazzuoccolo, Giuseppe Rajník, Jozef Tabarelli, Gloria |
| author_facet | Mattiolo, Davide Mazzuoccolo, Giuseppe Rajník, Jozef Tabarelli, Gloria |
| contents | A $d$-dimensional nowhere-zero $r$-flow on a graph $G$, an $(r,d)$-NZF from now on, is a flow where the value on each edge is an element of $\mathbb{R}^d$ whose (Euclidean) norm lies in the interval $[1, r-1]$. Such a notion is a natural generalization of the well-known concept of a circular nowhere-zero $r$-flow (i.e.\ $d = 1$). The minimum of the real numbers $r$ such that a graph $G$ admits an $(r, d)$-NZF is called the $d$-dimensional flow number of $G$ and is denoted by $ϕ_d(G)$. In this paper we provide a geometric description of some $d$-dimensional flows on a graph $G$, and we prove that the existence of a suitable cycle double cover of $G$ is equivalent, for $G$, to admit such a geometrically constructed $(r,d)$-NZF. This geometric approach allows us to provide upper bounds for $ϕ_{d-2}(G)$ and $ϕ_{d-1}(G)$, assuming that $G$ admits an (oriented) $d$-cycle double cover. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_19411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric description of $d$-dimensional flows of a graph Mattiolo, Davide Mazzuoccolo, Giuseppe Rajník, Jozef Tabarelli, Gloria Combinatorics 05C21 A $d$-dimensional nowhere-zero $r$-flow on a graph $G$, an $(r,d)$-NZF from now on, is a flow where the value on each edge is an element of $\mathbb{R}^d$ whose (Euclidean) norm lies in the interval $[1, r-1]$. Such a notion is a natural generalization of the well-known concept of a circular nowhere-zero $r$-flow (i.e.\ $d = 1$). The minimum of the real numbers $r$ such that a graph $G$ admits an $(r, d)$-NZF is called the $d$-dimensional flow number of $G$ and is denoted by $ϕ_d(G)$. In this paper we provide a geometric description of some $d$-dimensional flows on a graph $G$, and we prove that the existence of a suitable cycle double cover of $G$ is equivalent, for $G$, to admit such a geometrically constructed $(r,d)$-NZF. This geometric approach allows us to provide upper bounds for $ϕ_{d-2}(G)$ and $ϕ_{d-1}(G)$, assuming that $G$ admits an (oriented) $d$-cycle double cover. |
| title | Geometric description of $d$-dimensional flows of a graph |
| topic | Combinatorics 05C21 |
| url | https://arxiv.org/abs/2510.19411 |