Minimum Degree and Nowhere-Zero \(3\)-Flows

Abstract

Let \(G\) be a \(2\)-edge-connected simple graph on \(n\) vertices, \(n \geq 3\). It is known that if \(G\) satisfies \(d(x) \geq \frac{n}{2}\) for every vertex \(x \in V(G)\), then \(G\) has a nowhere-zero \(3\)-flow, with several exceptions.In this paper, we prove that, with ten exceptions, all graphs with at most two vertices of degree less than \(\frac{n}{2}\) have nowhere-zero \(3\)-flows. More precisely, if \(G\) is a \(2\)-edge-connected graph on \(n\) vertices, \(n \geq 3\), in which at most two vertices have degree less than \(\frac{n}{2}\), then \(G\)
has a nowhere-zero \(3\)-flow if and only if \(G\) is not one of ten completely described graphs.