Degree sum and nowhere-zero 3-flows

Let GG be a simple graph on nn vertices. In this paper, we prove that if GG satisfies the condition that d(x)+d(y)≥nd(x)+d(y)≥n for each xy∈E(G)xy∈E(G), then GG has no nowhere-zero 3-flow if and only if GG is either one of the five graphs on at most 6 vertices or one of a very special class of graphs on at least 6 vertices.