Ore condition and nowhere-zero 3-flows

Let G be a simple graph on n vertices, n >= 3. It is well known that if G satisfies the Ore condition that d(x) + d(y) >= 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 >= 3, in which d(x)+ d(y) >= 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.