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

We define a contravariant functor K from the category of finite graphs and graph morphisms to the category of finitely generated graded abelian groups and homomorphisms. For a graph X, an abelian group B, and a nonnegative integer j, an…

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

We study the flow spectrum ${\cal S}(G)$ and the integer flow spectrum $\overline{{\cal S}}(G)$ of signed $(2t+1)$-regular graphs. We show that if $r \in {\cal S}(G)$, then $r = 2+\frac{1}{t}$ or $r \geq 2 + \frac{2}{2t-1}$. Furthermore, $2…

A signed graph is a graph with a positive or negative sign on each edge. Regarding each edge as two half edges, an orientation of a signed graph is an assignment of a direction to each of its half edges such that the two half edges of a…

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that every regular graph of valency at least four admitting a solvable arc-transitive group of automorphisms admits a…

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…

Tutte's $3$-flow conjecture states that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. In this paper, we characterize all graphs with independence number at most $4$ that admit a nowhere-zero $3$-flow. The characterization…

Tutte initiated the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for every abelian group $\Gamma$ of order $n$, the number of nowhere-zero $\Gamma$-flows in…

Let $\Gamma$ be a graph, $A$ an abelian group, $\mathcal{D}$ a given orientation of $\Gamma$ and $R$ a unital subring of the endomorphism ring of $A$. It is shown that the set of all maps $\varphi$ from $E(\Gamma)$ to $A$ such that…

X. Hou, H.-J. Lai, P. Li and C.-Q. Zhang [J. Graph Theory 69 (2012) 464-470] showed that for a simple graph $G$ with $|V(G)|\ge 44$, if $\min\{\delta(G),\delta(G^c)\}\ge 4$, then either $G$ or its complementary graph $G^c$ has a…

Two well-known results in the world of nowhere-zero flows are Jaeger's 4-flow theorem asserting that every 4-edge-connected graph has a nowhere-zero $\mathbb{Z}_2 \times \mathbb{Z}_2$-flow and Seymour's 6-flow theorem asserting that every…

We investigate flows on graphs whose links have random capacities. For binary trees we derive the probability distribution for the maximal flow from the root to a leaf, and show that for infinite trees it vanishes beyond a certain threshold…

Let $r \geq 2$ be a real number. A complex nowhere-zero $r$-flow on a graph $G$ is an orientation of $G$ together with an assignment $\varphi\colon E(G)\to \mathbb{C}$ such that, for all $e \in E(G)$, the modulus of the complex number…

We prove that a signed graph admits a nowhere-zero $8$-flow provided that it is flow-admissible and the underlying graph admits a nowhere-zero $4$-flow. When combined with the 4-color theorem, this implies that every flow-admissible…

We generalise to signed graphs a classical result of Tutte [Canad. J. Math. 8 (1956), 13--28] stating that every integer flow can be expressed as a sum of characteristic flows of circuits. In our generalisation, the r\^ole of circuits is…

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…

Connectivity (or equivalently, unweighted maximum flow) is an important measure in graph theory and combinatorial optimization. Given a graph $G$ with vertices $s$ and $t$, the connectivity $\lambda(s,t)$ from $s$ to $t$ is defined to be…

In 1983, A. Bouchet extended W.T. Tutte's notion of nowhere-zero flows to signed graphs, and conjectured that every flow-admissible signed graph has a nowhere-zero 6-flow. In this paper we prove that every flow-admissible signed graph that…

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…

Let $G$ be a bridgeless cubic graph, and $\mu_2(G)$ the minimum number $k$ such that two 1-factors of $G$ intersect in $k$ edges. A cyclically $n$-edge-connected cubic graph $G$ has a nowhere-zero 5-flow if (1) $n \geq 6$ and $\mu_2(G) \leq…