Construction of graphs with given circular flow numbers

Abstract

AbstractSuppose r ≥ 2 is a real number. A proper r‐flow of a directed multi‐graph $\vec {G}=(V, E)$ is a mapping $f: E \to R$ such that (i) for every edge $e \in E$, $1 \leq |f(e)| \leq r-1$; (ii) for every vertex ${v} \in V$, $\sum _{e \in E^{+(v)}}f(e) - \sum _{e \in E^{-(v)}}f(e) = 0$. The circular flow number of a graph G is the least r for which an orientation of G admits a proper r‐flow. The well‐known 5‐flow conjecture is equivalent to the statement that every bridgeless graph has circular flow number at most 5. In this paper, we prove that for any rational number r between 2 and 5, there exists a graph G with circular flow number r. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 304–318, 2003