Flow-continuous mappings—The influence of the group

Abstract

Many conjectures at the core of graph theory can be formulated as questions about certain group-valued flows: examples include the cycle double-cover conjecture, the Berge–Fulkerson conjecture, and Tutte’s 3-flow, 4-flow, and 5-flow conjectures. As an approach to these problems, Jaeger, and DeVos, Nešetřil, 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 Z2, Z3, and Z.