Flow modules and nowhere-zero flows

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