# Circular flow number of regular class 1 graphs

 Importance: Medium ✭✭
 Author(s): Steffen, Eckhard
 Subject: Graph Theory » Coloring » » Nowhere-zero flows
 Keywords: nowhere-zero flow, edge-colorings, regular graphs
 Posted by: Eckhard Steffen on: August 5th, 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 . The circular flow number of is inf has a nowhere-zero -flow , and it is denoted by .

A graph with maximum vertex degree is a class 1 graph if its edge chromatic number is .

Conjecture   Let be an integer and a -regular graph. If is a class 1 graph, then .

The conjecture is true for , i.e. for cubic graphs. It says, that the circular flow number of -regular class 1 graphs is bounded by the circular flow number of the complete graph on vertices.

## Bibliography

[ES_2001] E. Steffen, Circular flow numbers of regular multigraphs, J. Graph Theory 36, 24 – 34 (2001)

*[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.