Related papers: Graph Puzzles II.1: Counterexamples to Jain's Seco…
We study the flow extension of graphs, i.e., pre-assigning a partial flow on the edges incident to a given vertex and aiming to extend to the entire graph. This is closely related to Tutte's $3$-flow conjecture(1972) that every…
The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection.…
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 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…
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…
In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero 6-flow. We verify this conjecture for the class of flow-admissible signed graphs possessing a spanning even Eulerian subgraph, which includes as 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…
An (r,alpha)-bounded excess flow ((r,alpha)-flow) in an orientation of a graph G=(V,E) is an assignment of a real "flow value" between 1 and r-1 to every edge. Rather than 0 as in an actual flow, some flow excess, which does not exceed…
Given a zero-sum function $\beta : V(G) \rightarrow \mathbb{Z}_3$ with $\sum_{v\in V(G)}\beta(v)=0$, an orientation $D$ of $G$ with $d^+_D(v)-d^-_D(v)= \beta(v)$ in $\mathbb{Z}_3$ for every vertex $v\in V(G)$ is called a…
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…
A connected graph G is 3-flow-critical if G does not have a nowhere-zero 3-flow, but every proper contraction of G does. We prove that every n-vertex 3-flow-critical graph other than K_2 and K_4 has at least 5n/3 edges. This bound is tight…
Bouchet conjectured in 1983 that every flow-admissible signed graph admits a nowhere-zero 6-flow which is equivalent to the restriction to cubic signed graphs. In this paper, we proved that every flow-admissible $3$-edge-colorable cubic…
The number of nowhere zero Z_Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh that…
For integers $a\ge 2b>0$, a \emph{circular $a/b$-flow} is a flow that takes values from $\{\pm b, \pm(b+1), \dots, \pm(a-b)\}$. The Planar Circular Flow Conjecture states that every $2k$-edge-connected planar graph admits a circular…
A function $f:N\rightarrow N$ is sublinear, if \[\lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0.\] If $A$ is an Abelian group, $G$ is a graph and $\phi$ is an $A$-flow in $G$, then let $N(\phi)$ be the nullity of $\phi$, that is, the set of…
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…
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…
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…
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.
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…