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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
Recomm. for undergrads: no
Posted by: Eckhard Steffen
on: August 5th, 2015

A nowhere-zero $ r $-flow $ (D(G),\phi) $ on $ G $ is an orientation $ D $ of $ G $ together with a function $ \phi $ from the edge set of $ G $ into the real numbers such that $ 1 \leq |\phi(e)| \leq r-1 $, for all $ e \in E(G) $, and $ \sum_{e \in E^+(v)}\phi(e) = \sum_{e \in E^-(v)}\phi(e), \textrm{ for all } v \in V(G) $. The circular flow number of $ G $ is inf$ \{ r | G $ has a nowhere-zero $ r $-flow $ \} $, and it is denoted by $ F_c(G) $.

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

Conjecture   Let $ t \geq 1 $ be an integer and $ G $ a $ (2t+1) $-regular graph. If $ G $ is a class 1 graph, then $ F_c(G) \leq 2 + \frac{2}{t} $.

The conjecture is true for $ t=1 $, i.e. for cubic graphs. It says, that the circular flow number of $ (2t+1) $-regular class 1 graphs is bounded by the circular flow number of the complete graph on $ 2t+2 $ vertices.

Related problems

(2 + epsilon)-flow conjecture
Jaeger's modular orientation conjecture

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.
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