# Circular flow numbers of $r$-graphs

 Importance: Medium ✭✭
 Author(s): Steffen, Eckhard
 Subject: Graph Theory
 Keywords: flow conjectures nowhere-zero flows
 Posted by: Eckhard Steffen on: August 6th, 2015

A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and .

A -regular graph is a -graph if for every with odd.

Conjecture   Let be an integer. If is a -graph, then .

Since every -regular class 1 graph is a -graph, the truth of this conjecture would imply the truth of the conjecture on the circular flow number of regular class 1 graphs. If it is true for even , say , then Jaeger's modular orientation conjecture is true for -regular graphs and hence, by a result of Jaeger, it would imply the truth of Tutte's 5-flow conjecture. For it is Tutte's 3-flow conjecture.

## Bibliography

*[ES_2015]E. Steffen, Edge-colorings and circular flow numbers on regular graphs, J. Graph Theory 79, 1–7, 2015

* indicates original appearance(s) of problem.