Related papers: Nowhere-Zero $\vec k$-Flows on Graphs

A nowhere-zero $k$-flow on a graph $\Gamma$ is a mapping from the edges of $\Gamma$ to the set $\{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ$ such that, in any fixed orientation of $\Gamma$, at each node the sum of the labels over the edges…

In this article we introduce the flow polynomial of a digraph and use it to study nowhere-zero flows from a commutative algebraic perspective. Using Hilbert's Nullstellensatz, we establish a relation between nowhere-zero flows and dual…

Given a graph $G$, the number of nowhere-zero $\ZZ_q$-flows $\phi_G(q)$ is known to be a polynomial in $q$. We extend the definition of nowhere-zero $\ZZ_q$-flows to simplicial complexes $\Delta$ of dimension greater than one, and prove the…

Given an oriented graph G, the modular flow polynomial counts the number of nowhere-zero Z_k-flows of G. We give a description of the modular flow polynomial in terms of (open) Ehrhart polynomials of lattice polytopes. Using…

Kochol introduced the assigning polynomial $F(G,\alpha;k)$ to count nowhere-zero $(A,b)$-flows of a graph $G$, where $A$ is a finite Abelian group and $\alpha$ is a $\{0,1\}$-assigning from a family $\Lambda(G)$ of certain nonempty vertex…

A $3$-dimensional nowhere-zero flow on a graph $G$ is a flow where each edge is assigned a $3$-dimensional vector with unit norm (which corresponds to the points of a $2$-dimensional unit sphere $S^2$). K. Jain posed two conjectures related…

The study of nowhere-zero flows began with a key observation of Tutte that in planar graphs, nowhere-zero k-flows are dual to k-colourings (in the form of k-tensions). Tutte conjectured that every graph without a cut-edge has a nowhere-zero…

For a bridgeless graph $G$, its flow polynomial is defined to be the function $F(G,q)$ which counts the number of nonwhere-zero $\Gamma$-flows on an orientation of $G$ whenever $q$ is a positive integer and $\Gamma$ is an additive Abelian…

We prove that every regular graph of valency at least four whose automorphism group contains a nilpotent subgroup acting transitively on the vertex set admits a nowhere-zero 3-flow.

We give a compact variation of Seymour's proof that every $2$-edge-connected graph has a nowhere-zero $\mathbb{Z}_2 \times \mathbb{Z}_3$-flow.

Let $G$ be a graph. A zero-sum flow of $G$ is an assignment of non-zero real numbers to the edges of $G$ such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be a natural number. A zero-sum $k$-flow is a…

In contrast to ordinary graphs, the number of the nowhere-zero group-flows in a signed graph may vary with different groups, even if the groups have the same order. In fact, for a signed graph $G$ and non-negative integer $d$, it was shown…

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. These results may be seen as a counting alternative to the well-known proofs of existence of $Z_3$-, $Z_4$-, and $Z_6$-flows. In the dual setting,…

We survey known results related to nowhere-zero flows and related topics, such as circuit covers and the structure of circuits of signed graphs. We include an overview of several different definitions of signed graph colouring.

In 1983, Bouchet proved that every bidirected graph with a nowhere-zero integer-flow has a nowhere-zero 216-flow, and conjectured that 216 could be replaced with 6. This paper shows that for cyclically 5-edge-connected bidirected graphs…

We initiate the study of nowhere-zero flow reconfiguration. The natural question is whether any two nowhere-zero $k$-flows of a given graph $G$ are connected by a sequence of nowhere-zero $k$-flows of $G$, such that any two consecutive…

Many basic properties in Tutte's flow theory for unsigned graphs do not have their counterparts for signed graphs. However, signed graphs without long barbells in many ways behave like unsigned graphs from the point view of flows. In this…

Bouchet conjectured in 1983 that each signed graph that admits a nowhere-zero flow has a nowhere-zero 6-flow. We prove that the conjecture is true for all signed series-parallel graphs. Unlike the unsigned case, the restriction to…

This paper is devoted to a detailed study of nowhere-zero flows on signed eulerian graphs. We generalise the well-known fact about the existence of nowhere-zero $2$-flows in eulerian graphs by proving that every signed eulerian graph that…

A circular nowhere-zero $r$-flow on a bridgeless graph $G$ is an orientation of the edges and an assignment of real values from $[1, r-1]$ to the edges in such a way that the sum of incoming values equals the sum of outgoing values for…