Related papers: Many flows in the group connectivity setting

If $X=(V(X),E(X))$ and $Y=(V(Y),E(Y))$ are $n$-vertex graphs, then their friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertices are the bijections from $V(X)$ to $V(Y)$ in which two bijections $\sigma$ and $\sigma'$ are…

Given a 2-generated finite group $G$, the non-generating graph of $G$ has as vertices the elements of $G$ and two vertices are adjacent if and only if they are distinct and do not generate $G$. We consider the graph $\Sigma(G)$ obtained…

The flow semigroup, introduced by John Rhodes, is an invariant for digraphs and a complete invariant for graphs. After collecting together previous partial results, we refine and prove Rhodes's conjecture on the structure of the maximal…

The Bach flow is a fourth order geometric flow defined on four manifolds. For a compact manifold, it is a conformally modified gradient flow for the $L^2$-norm of the Weyl curvature. In this paper we study the Bach flow on four-dimensional…

Flow-based methods have achieved significant success in various generative modeling tasks, capturing nuanced details within complex data distributions. However, few existing works have exploited this unique capability to resolve…

For a group $\Gamma$, a $\Gamma$-labelled graph is an undirected graph $G$ where every orientation of an edge is assigned an element of $\Gamma$ so that opposite orientations of the same edge are assigned inverse elements. A path in $G$ is…

A graph is called strongly $\Z_{2k+1}$-connected if for each boundary function $\beta: V(G)\mapsto \Z_{2k+1}$ with $\sum_{v\in V(G)}\beta(v)\equiv 0\pmod{2k+1}$, there exists an orientation $D$ of $G$ such that $d_D^+(v) - d_D^-(v) \equiv…

This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R^2. We show that any such flow is a shear flow, that is, it is parallel to…

The zero forcing number of a simple graph, written $Z(G)$, is a NP-hard graph invariant which is the result of the zero forcing color change rule. This graph invariant has been heavily studied by linear algebraists, physicists, and graph…

Nash-Williams proved in 1960 that a finite graph admits a $k$-arc-connected orientation if and only if it is $2k$-edge-connected, and conjectured that the same result should hold for all infinite graphs, too. Progress on Nash-Williams's…

It is known that, if we take a countable model of Zermelo--Fraenkel set theory ZFC and "undirect" the membership relation (that is, make a graph by joining $x$ to $y$ if either $x\in y$ or $y\in x$), we obtain the Erd\H{o}s--R\'enyi random…

We consider the problem of multicommodity flows in planar graphs. Seymour showed that if the union of supply and demand graphs is planar, then the cut condition is sufficient for routing demands. Okamura-Seymour showed that if all demands…

Let $\mathbb{H}$ be the sub-Riemannian Heisenberg group. That $\mathbb{H}$ supports a rich family of quasiconformal mappings was demonstrated by Kor\'{a}nyi and Reimann using the so-called flow method. Here we supply further evidence of the…

Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or without boundary, for which there exists a steady flow among…

A mapping $l : E(G) \rightarrow A$, where $A$ is an abelian group which written additively, is called a labeling of the graph $G$. For every positive integer $h \geqslant 2$, a graph $G$ is said to be zero-sum $h$-magic if there is an edge…

The analogue of Hadwiger's conjecture for the immersion order states that every graph $G$ contains $K_{\chi (G)}$ as an immersion. If true, it would imply that every graph with $n$ vertices and independence number $\alpha$ contains…

In 1982 V\'{i}ctor Neumann-Lara introduced the dichromatic number of a digraph $D$ as the smallest integer $k$ such that the vertices $V$ of $D$ can be colored with $k$ colors and each color class induces an acyclic digraph. Later a flow…

We prove that for every countable discrete group $G$, there is a $G$-flow on $\omega^*$ that has every $G$-flow of weight $\leq\! \aleph_1$ as a quotient. It follows that, under the Continuum Hypothesis, there is a universal $G$-flow of…

This paper studies signed graphs with possible outer-edges. We introduce and investigate the chain group, the boundary operator, the co-boundary operator, the flow group, the tension group, the homology group, the cohomology group, with…

A graph $G$ is $\mathcal S_3$-connected if, for any mapping $\beta : V (G) \mapsto {\mathbb Z}_3$ with $\sum_{v\in V(G)} \beta(v)\equiv 0\pmod3$, there exists a strongly connected orientation $D$ satisfying $d^{+}_D(v)-d^{-}_D(v)\equiv…