Flow-continuous mappings -- influence of the group

TL;DR

This paper explores how the structure of the group influences flow-continuous mappings in graph theory, revealing that the number of such mappings depends only on the group's exponent and identifying key groups for cubic graphs.

Contribution

It proves that the count of flow-continuous mappings depends solely on the group's exponent and characterizes the groups relevant for cubic graphs.

Findings

Number of flow-continuous mappings depends only on the group's exponent.

For cubic graphs, only groups Z2, Z3, and Z are relevant.

An algebraic structure describes when a mapping is flow-continuous.

Abstract

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 $\Z$.