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