Related papers: On 1-sum flows in undirected graphs
We study the resilience of random and pseudorandom directed graphs with respect to the property of having long directed cycles. For every $0 < \gamma < 1/2$ we find a constant $c=c(\gamma)$ such that the following holds. Let $G=(V,E)$ be a…
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…
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…
Tutte's 5-Flow Conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. In 2004, Kochol proved that the conjecture is equivalent to its restriction on cyclically 6-edge connected cubic graphs. We prove that every…
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…
Flows over time are used to model many real-world logistic and routing problems. The networks underlying such problems -- streets, tracks, etc. -- are inherently undirected and directions are only imposed on them to reduce the danger of…
In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero $6$-flow. Bouchet himself proved that such signed graphs admit nowhere-zero $216$-flows and Zyka further proved that such signed graphs…
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…
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 $\gamma$-graph of a graph $G$ is the graph whose vertices are labelled by the minimum dominating sets of $G$, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size $\gamma(G)$) intersect in a…
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 transshipment problem (G, d, \lambda) is modeled by a directed graph G = (V, E) with weighted vertices d = (d_v | v \in V) and directed edges \lambda = (\lambda_e | e \in E) interpreted as follows: G is a communication or transportation…
We study the problem of enumerating the $k$-arc-connected orientations of a graph $G$, i.e., generating each exactly once. A first algorithm using submodular flow optimization is easy to state, but intricate to implement. In a second…
We consider AF-flows, i.e., one-parameter automorphism groups of a unital simple C*-algebra which leave invariant the dense union of an increasing sequence of finite-dimensional *-subalgebras, and derive two properties for these; an absence…
A pair of graphs $(\Gamma,\Sigma)$ is called unstable if their direct product $\Gamma\times\Sigma$ admits automorphisms not from $\mathrm{Aut}(\Gamma)\times\mathrm{Aut}(\Sigma)$, and such automorphisms are said to be unexpected. The…
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 $G=(V,E)$ and a set $S_0\subseteq V$, an irreversible $k$-threshold conversion process on $G$ is an iterative process wherein, for each $t=1,2,\dots$, $S_t$ is obtained from $S_{t-1}$ by adjoining all vertices that have at…
The notion of smoothness was introduced originally in the context of step systems on connected graphs. Smoothness turns out to be a very general property of metrics defined by a five-point condition. Restricted to graphs, it is closely…
We introduce a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a…
Inspired by a famous characterization of perfect graphs due to Lov\'{a}sz, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $\alpha(H) + \omega(H) \geq |V(H)|$. (Here $\alpha$ and $\omega$ denote the…