Flow modules and nowhere-zero flows

Abstract

Let $$\varGamma $$ be a graph, A an abelian group, $${\mathcal {D}}$$ a given orientation of $$\varGamma $$ and R a unital subring of the endomorphism ring of A. It is shown that the set of all mappings $$\varphi $$ from $$E(\varGamma )$$ to A such that $$({\mathcal {D}},\varphi )$$ is an A-flow forms a left R-module. Let $$\varGamma $$ be a union of two subgraphs $$\varGamma _{1}$$ and $$\varGamma _{2}$$ , and $$p^n$$ a prime power. It is proved that $$\varGamma $$ admits a nowhere-zero $$p^n$$ -flow if $$\varGamma _{1}$$ and $$\varGamma _{2}$$ have at most $$p^n-2$$ common edges and both admit nowhere-zero $$p^n$$ -flows. Moreover, it is proved that $$\varGamma $$ admits a nowhere-zero 4-flow if $$\varGamma _{1}$$ and $$\varGamma _{2}$$ both have nowhere-zero 4-flows and their common edges induce a subgraph of size at most 2 or a connected subgraph of size 3. This result can be seen as a generalization of a theorem of Catlin that a graph admits a nowhere-zero 4-flow if it is a union of a cycle of length at most 4 and a subgraph admitting a nowhere-zero 4-flow.