Ore Condition and Nowhere-Zero 3-Flows

Abstract

Let $G$ be a simple graph on $n$ vertices, $n\geq 3$. It is well known that if $G$ satisfies the Ore condition that $d(x)+d(y)\geq n$ for every pair of nonadjacent vertices $x$ and $y$, then $G$ has a Hamiltonian circuit, which implies that $G$ has a nowhere-zero 4-flow. But it is not necessary for $G$ to have a nowhere-zero 3-flow. In this paper, we prove that with six exceptions, all graphs satisfying the Ore condition have a nowhere-zero 3-flow. More precisely, if $G$ is a graph on $n$ vertices, $n\geq 3$, in which $d(x)+d(y)\geq n$ for every pair of nonadjacent vertices $x$ and $y$, then $G$ has no nowhere-zero 3-flow if and only if $G$ is one of six completely described graphs.