Flow modules and nowhere-zero flows

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 $\phi$ from $E(\Gamma )$ to $A$ such that $(\mathcal{D},\phi )$ 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.