Integral Flows on Bidirected Eulerian Graphs

Abstract

It is proved that every flow-admissible bidirected eulerian graph has a nowhere-zero 4-flow $f$ and a circle $C$, where either $C$ is unbalanced or $|E(C)|\geq 3$, such that $|f(e)|=3$ only if $e\in E(C)$. It is also proved, with an interesting new method, that in a flow-admissible connected bidirected graph, the support of any ${\bf Z}_2$-flow is contained in the support of a 5-flow.