Related papers: Nowhere-zero 8-flows in cyclically 5-edge-connecte…
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…
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's 3-flow conjecture asserts that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow…
In 1981 Seymour proved his famous 6-flow theorem asserting that every 2-edge-connected graph has a nowhere-zero flow in the group ${\mathbb Z}_2 \times {\mathbb Z}_3$ (in fact, he offers two proofs of this result). In this note we give a…
Tutte's $3$-flow conjecture says that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. Kochol (2001) showed that it is enough to prove this conjecture for $5$-edge-connected graphs. Former, Jaeger, Linial, Payan, and Tarsi…
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…
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…
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,…
Lov\'{a}sz et al. proved that every $6$-edge-connected graph has a nowhere-zero $3$-flow. In fact, they proved a more technical statement which says that there exists a nowhere zero $3$-flow that extends the flow prescribed on the incident…
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…
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…
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.
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…
A set $R\subseteq E(G)$ of a graph $G$ is $k$-removable if $G-R$ has a nowhere-zero $k$-flow. We prove that every graph $G$ admitting a nowhere-zero $4$-flow has a $3$-removable subset consisting of at most $\frac{1}{6}|E(G)|$ edges. This…
A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of $G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at most $k-2$…
We investigate multidimensional nowhere-zero flows of bridgeless graphs. By extending the established use of the Euclidean norm, this paper considers the Manhattan and Chebyshev norms, leading to the definition of the flow numbers…
The presented paper studies the flow number $F(G,\sigma)$ of flow-admissible signed graphs $(G,\sigma)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\sigma)$ there is a set…
We introduce and study a multivariate function that counts nowhere-zero flows on a graph G, in which each edge of G has an individual capacity. We prove that the associated counting function is a piecewise-defined polynomial in these…
In 1972 Tutte famously conjectured that every 4-edge-connected graph has a nowhere zero 3-flow; this is known to be equivalent to every 5-regular, 4-edge-connected graph having an edge orientation in which every in-degree is either 1 or 4.…
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.