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Many questions at the core of graph theory can be formulated as questionsabout certain group-valued flows: examples are the cycle double coverconjecture, Berge-Fulkerson conjecture, and Tutte's 3-flow, 4-flow, and 5-flowconjectures. As an approach to these problems Jaeger and DeVos,Ne\v{s}et\v{r}il, and Raspaud define a notion of graph morphisms continuouswith respect to group-valued flows. We discuss the influence of the group onthese maps. In particular, we prove that the number of flow-continuous mappingsbetween two graphs does not depend on the group, but only on the largest orderof an element of the group (i.e., on the exponent of the group). Further, thereis a nice algebraic structure describing for which groups a mapping isflow-continuous.  On the combinatorial side, we show that for cubic graphs the only relevantgroups are $\Z_2$, $\Z_3$, and $\Z$.

Flow-continuous mappings -- influence of the group