Algebraic Flow Theory of Infinite Graphs

arXiv: Combinatorics, 2020

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 2-edge-connected graph has a nowhere-zero $\mathbb{Z}_6$-flow. Dvořak and the last two authors of this paper extended these results by proving the existence of exponentially many nowhere-zero flows under the same assumptions. We revisit this setting and provide extensions and simpler proofs of these results. The concept of a nowhere-zero flow was extended in a significant paper of Jaeger, Linial, Payan, and Tarsi to a choosability-type setting. For a fixed abelian group $\Gamma$, an oriented graph $G = (V,E)$ is called $\Gamma$-connected if for every function $f : E \rightarrow \Gamma$ there is a flow $\phi : E \rightarrow \Gamma$ with $\phi(e) \neq f(e)$ for every $e \in E$ (note that taking $f = 0$ forces $\phi$ to be nowhere-zero). Jaege...

European Journal of Combinatorics, 2014

We exhibit explicit constructions of contractors for the graph parameter counting the number of B-flows of a graph, where B is a subset of a finite Abelian group closed under inverses. These constructions are of great interest because of their relevance to the family of B-flow conjectures formulated by Tutte, Fulkerson, Jaeger, and others.

Electronic Journal of Linear Algebra, 2016

Let $G=(V, E)$ be a simple undirected graph. For a given set $L\subset \mathbb{R}$, a function $\omega: E \longrightarrow L$ is called an $L$-flow. Given a vector $\gamma \in \mathbb{R}^V$, $\omega$ is a $\gamma$-$L$-flow if for each $v\in V$, the sum of the values on the edges incident to $v$ is $\gamma(v)$. If $\gamma(v)=c$, for all $v\in V$, then the $\gamma$-$L$-flow is called a $c$-sum $L$-flow. In this paper, the existence of $\gamma$-$L$-flows for various choices of sets $L$ of real numbers is studied, with an emphasis on 1-sum flows. Let $L$ be a subset of real numbers containing $0$ and denote $L^*:=L\setminus \{0\}$. Answering a question from S. Akbari, M. Kano, and S. Zare. A generalization of $0$-sum flows in graphs. \emph{Linear Algebra Appl.}, 438:3629--3634, 2013.], the bipartite graphs which admit a $1$-sum $\mathbb{R}^*$-flow or a $1$-sum $\mathbb{Z}^*$-flow are characterized. It is also shown that every $k$-regular graph, with $k$ either odd or congruent to 2 modul...

Czechoslovak Mathematical Journal, 2005

Journal of Graph Theory, 2003

It is shown that the Cartesian product of two nontrivial connected graphs admits a nowhere-zero 4-flow. If both factors are bipartite, then the product admits a nowhere-zero 3-flow.

viXra, 2018

The main purpose of this paper is to extend Banach or Hilbert spaces to Banach or Hilbert continuity flow spaces over topological graphs and establish differentials on continuity flows for characterizing their globally change rate.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Journal of Pure and Applied Algebra, 2012

The algebraic entropy h, defined for endomorphisms φ of abelian groups G, measures the growth of the trajectories of non-empty finite subsets F of G with respect to φ. We show that this growth can be either polynomial or exponential. The greatest φ-invariant subgroup of G where this growth is polynomial coincides with the greatest φ-invariant subgroup P(G, φ) of G (named Pinsker subgroup of φ) such that h(φ ↾ P(G,φ) ) = 0. We obtain also an alternative characterization of P(G, φ) from the point of view of the quasi-periodic points of φ.

2012

This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of S∞. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows us to provide a partial answer to a question posed in [KPT05] by Kechris, Pestov and Todorcevic.

Journal of Graph Theory, 2011

A shortest cycle cover of a graph G is a family of cycles which together cover all the edges of G and the sum of their lengths is minimum. In this article we present upper bounds to the length of shortest cycle covers, associated with the existence of two types of nowhere-zero