Flow modules and nowhere-zero flows

TL;DR

This paper investigates the algebraic structure of flows in graphs and provides new conditions under which graphs admit nowhere-zero flows, including specific results for 4-flows and prime power flows.

Contribution

It introduces a module structure for flow sets and establishes new sufficient conditions for the existence of nowhere-zero flows in graph unions.

Findings

Flow sets form a left R-module.

Graphs with certain subgraph properties admit nowhere-zero p^n-flows.

New conditions for 4-flows extend previous results.

Abstract

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 $(\mathcal{D},\varphi)$ is an $A$-flow forms a left $R$-module. Let $\Gamma$ be a union of two subgraphs $\Gamma_{1}$ and $\Gamma_{2}$, and $p^n$ a prime power. It is proved that $\Gamma$ admits a nowhere-zero $p^n$-flow if $\Gamma_{1}$ and $\Gamma_{2}$ have at most $p^n-2$ common edges and both have nowhere-zero $p^n$-flows. More important, it is proved that $\Gamma$ admits a nowhere-zero $4$-flow if $\Gamma_{1}$ and $\Gamma_{2}$ both have nowhere-zero $4$-flows and their common edges induce a connected subgraph of $\Gamma$ of size at most $3$. This covers a result of Catlin that a graph admits a nowhere-zero $4$-flow if it is a union of a $4$-cycle and a subgraph admiting a nowhere-zero $4$-flow.