# 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 $r$-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 $k$.

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

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

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.

## Bibliography

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[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.