2025/10/22 by Davide Mattiolo, Mattiolo, Davide, Giuseppe Mazzuoccolo +5
Computer Science · Mathematics · #05C21 #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2510.19411
openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
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 ℝd 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 ϕd(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 ϕd-2(G) and ϕd-1(G), assuming that G admits an (oriented) d-cycle double cover.