X. Hou, H.-J. Lai, P. Li and C.-Q. Zhang [J. Graph Theory 69 (2012) 464-470] showed that for a simple graph $G$ with $|V(G)|\ge 44$, if $\min\{\delta(G),\delta(G^c)\}\ge 4$, then either $G$ or its complementary graph $G^c$ has a nowhere-zero $3$-flow. In this paper, we improve this result by showing that if $|V(G)|\ge 32$ and $\min\{\delta(G),\delta(G^c)\}\ge 4$, then either $G$ or $G^c$ has flow index strictly less than $3$. Our result is proved by a newly developed closure operation and contraction method.

中文翻译：

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流量小于三的互补图

X. 侯，H.-J。Lai，P. Li 和 C.-Q。张[J. 图论 69 (2012) 464-470] 表明对于一个简单的图 $G$ 和 $|V(G)|\ge 44$，如果 $\min\{\delta(G),\delta(G^c )\}\ge 4$，则 $G$ 或其互补图 $G^c$ 具有无处零 $3$-flow。在本文中，我们通过证明如果 $|V(G)|\ge 32$ 和 $\min\{\delta(G),\delta(G^c)\}\ge 4$ 来改进这个结果，那么$G$ 或 $G^c$ 的流量指数严格小于 $3$。我们的结果由新开发的闭合操作和收缩方法证明。